CCN Math Review Chapter : Functions Section.1: Functions.1.1: How functions are used.1.: Methods for defining functions.1.3: The graph of a function.1.: Domain and range.1.5: Relations, functions, and graphs.1.6: Examples of graphs of functions.1.7: The domain really matters!.1.8: Quiz review Section.: Graphs..1: Graphing a real-life function..: The effect of the choice of domain..3: Special features of graphs..: Linear functions defined on intervals..5: Graphing a polynomial on an interval..6: Piecewise defined functions..7: A three-part piecewise linear graph..8: The vertical line test..9: Quiz review Section.3: Analyzing graphs.3.1: Maximum and minimum points.3.: Function increasing or decreasing.3.3: Given h(t), find t.3.: A degree 3 polynomial.3.5: Count solutions of h(t) = K.3.6: Quiz review Section.: Rate of change..1: Average rate of change..: as the slope of a line..3: Speedometers display average velocity..: over a small time interval..5: Quiz review Section.5: Transformations.5.1: Transforming equations.5.: Six ways to transform graphs.5.3: Comparing transformations.5.: More examples.5.5: Quiz review Section.6 : Function composition.6.1: Combining functions.6.: Composing functions.6.3: Parenthesis rule for functions.6.: Inverse functions undo each other.6.5: Quiz review Section.7 :Inverse functions.7.1: Formulas and functions.7.: Composing f and f 1.7.3: The horizontal line test.7.: Restricting the domain.7.5: Graphing f and f 1 on the same grid.7.6: Quiz review All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Chapter : Functions 8/9/16 Frame 1
Section.1: What is a function?.1.1: How functions are used.1.: Methods for defining functions.1.3: The graph of a function.1.: Domain and range.1.5: Relations, functions, and graphs.1.6: Examples of graphs of functions.1.7: Why you specify the domain.1.8: Quiz review All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame
.1.1 Functions can be used to describe how one quantity depends on another A quantity is a number together with a unit of measurement, such as feet or seconds. One quantity of interest often depends on another. Classical physics assumes that the position of a moving object depends on time. Other examples include: The area of a square depends on the length of its side. The price of a shopping bag full of apples depends on how many pounds of apples are in it. The position of a missile depends on the time that has elapsed since it was launched. Functions are often used to describe how one quantity depends on another. I drop a ball off the top of a 1-foot building. From the time the ball is dropped until it hits the ground, the ball s height (measured in feet) above the ground at time t seconds is 1 16t. The ball s height above the ground depends on how long it has been falling. Use the symbol h(t) to abbreviate the ball s height in feet at time t seconds after it is dropped. Then h(t) = 1 16t from time t = until the ball hits the ground. Example 1 : How high above the ground is the ball after seconds? Solution: To find h(), substitute for t in h(t) = 1 16t. Then h() = 1 16() = 1 16() = 1 6 = 36 Answer: After seconds, the ball is 36 feet above the ground. Example : When does the ball hit the ground? Solution: Since h(t) is the ball s height at time t, and the ball s height when it hits the ground is, solve h(t) = for t. h(t) = 1 16t = 16t = 1 t = 1 16 = 5 t = ± 5 = ± 5 = ± 5 and so t =.5 or t =.5. Since we require t, the only valid answer is t =.5. Answer: The ball hits the ground.5 seconds after it is dropped. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 3
We talked earlier about functions that are specified by a formula called a function definition. But we never stated what the word function means. Textbook definition of function A function f is a rule that assigns to each element x in a set A exactly one element, f(x), in a set B. In these notes A and B are sets of real numbers. So a function starts with a real number input x and produces a real number output f(x). The statement that a function is a rule is a bit misleading. It suggests that different rules for getting from the input to the output might define different functions. That s not so. The only thing a function cares about is: for a given input, what is the output? It s reasonable to say that the rules f(x) = x 1 and g(x) = (x 1)(x + 1) are different: they require different calculations to find their respective outputs. However, they define exactly the same function, because f(x) = g(x) for every possible input x. Now we can say exactly what a function is. A function is a set of ordered pairs An ordered pair [of real numbers] is (x, y), where x and y are real numbers. The ordered pair s input is x and its output is y. A function F is any set of ordered pairs of real numbers in which no input has two different outputs. If x is an input, then F (x) is the output. The domain of F consists of all its inputs. The range of F consists of all its outputs. The function F is the set of all ordered pairs (x, F (x)), where x is an input for F. To describe a function by using a formula, you must state the domain explicitly. Functions with different domains are different, even if they are described by the same rule. How to describe or represent a function f If the number of ordered pairs is infinite, state a rule or formula for f(x) for inputs x in a specified domain, or try to plot the ordered pairs as points (x, f(x)) in the x,y-plane. If the number of ordered pairs is finite, you may also list the ordered pairs as a set, or list the ordered pairs in a table. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame
.1. Methods for defining functions A function defined as a finite set: The set F = {(, ), (1, 3), (, ), (3, 5), (, 6)} is a function because each input has only one output. For this function, F () =, F (1) = 3, F () =, F (3) = 5, and F () = 6. The domain of F consists of the five real numbers {, 1,, 3, }. The set G = {(, ), (1, 3), (, ), (3, 5), (1, 6)} is NOT a function. since the pairs (1, 3) and (1, 6) have the same input but different outputs. Functions consisting of finitely many pairs are useful in discussions of probability. However, most functions used in calculus and science consist of infinitely many pairs and must be specified by a formula called a function definition. Examples of functions defined by a formula: The function with domain all real numbers and formula f(x) = x 1 consists of all number pairs (x, x 1) where x is any real number. The function with domain all real numbers and formula g(x) = (x + 1)(x 1) consists of all number pairs (x, (x + 1)(x 1)), where x is any real number. In this example, f and g are the same function, even though their formulas are different. That s because they have the same domain and f(x) = g(x) for all x in that domain. In general, a function formula should specify the domain of allowed inputs. If it does not, use the function s natural domain, which consists of all real number inputs x for which the formula makes sense. How to find the natural domain of a function f(x) Find all values of x for which f(x) involves a fraction with denominator. Find all values of x for which f(x) involves an even root of a negative number. The natural domain of f consists of all real numbers x other than those found in the previous two bullets. Example: h(x) = x defines a function with natural domain [, ). That s because the square root of a negative real number is not a real number. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 5
.1.3 The graph of a function. What is the graph of a function? The graph of a function F consists of all points obtained by plotting number pairs (x, F (x)), where x is in the domain of the function. In this course, most functions have infinite domains: either all real numbers or an interval of real numbers such as [3, 7]. Here lists or tables won t work: such functions are described by formulas or graphs. To draw the graph of a function, draw dots on the coordinate plane, one for each ordered pair (x, F (x)). The number of dots plotted, even by a computer, is finite. But one can try to get a reasonable representation by connecting the dots. This must be done with care, as will be shown later. Example 3. The function F with domain {, 1,, 3, } is defined by F () =, F (1) = 3, F () =, F (3) = 5, F () = 6. Describe F using four different methods. Solution: As a set F = {(, ), (1, 3), (, ), (3, 5), (, 6)} as a table with inputs in column 1 and outputs in column : x F (x) 1 3 3 5 6 as a function definition specifying both domain and formula: F (x) = x + with domain {, 1,, 3, } as a graph, obtained by plotting the points (, ), (1, 3), (, ), (3, 5), and (, 6). 6-1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 6
.1. Domain and range Example : Find the natural domain of the function 1 f(x) = x+5. Solution: The square root of a number less than is not defined. Thus we need x + 5, which means x 5. Division by zero is undefined. Thus we require x + 5 and so x 5. Answer: Combine the above requirements. The domain of f is x > 5. In interval notation: ( 5, ). Example 5: 1 Using interval notation, find the natural domain of x 3x+ Solution: x can be any number that makes the denominator non-zero. To find the exceptions, solve x 3x + = (x )(x 1) = x = ; x = 1. Thus x can be any number other than 1 or. Answer: The domain of f is (, 1) (1, ) (, ). Example: Below is a graph given by formula F (x) = 5x. Keep on clicking to find the domain 3 and range of the function sketched. To find the domain, slide all points on the graph down to the x-axis. The domain of f is [1, ). To find the range, slide all points on the graph left to the y-axis. The range of f is [1, 6). ; (,6) How to find domain and range of function f from its graph The domain of f consists of all points on the horizontal axis that lie on vertical lines through points on the graph. The range of f consists of all points on the vertical axis that lie on horizontal lines through points on the graph. (1,1)
.1. Domain and range Example : Find the natural domain of the function 1 f(x) = x+5. Solution: The square root of a number less than is not defined. Thus we need x + 5, which means x 5. Division by zero is undefined. Thus we require x + 5 and so x 5. Answer: Combine the above requirements. The domain of f is x > 5. In interval notation: ( 5, ). Example 5: 1 Using interval notation, find the natural domain of x 3x+ Solution: x can be any number that makes the denominator non-zero. To find the exceptions, solve x 3x + = (x )(x 1) = x = ; x = 1. Thus x can be any number other than 1 or. Answer: The domain of f is (, 1) (1, ) (, ). Example: Below is a graph given by formula F (x) = 5x. Keep on clicking to find the domain 3 and range of the function sketched. To find the domain, slide all points on the graph down to the x-axis. The domain of f is [1, ). To find the range, slide all points on the graph left to the y-axis. The range of f is [1, 6). ; (,6) How to find domain and range of function f from its graph The domain of f consists of all points on the horizontal axis that lie on vertical lines through points on the graph. The range of f consists of all points on the vertical axis that lie on horizontal lines through points on the graph. (1,1)
.1. Domain and range Example : Find the natural domain of the function 1 f(x) = x+5. Solution: The square root of a number less than is not defined. Thus we need x + 5, which means x 5. Division by zero is undefined. Thus we require x + 5 and so x 5. Answer: Combine the above requirements. The domain of f is x > 5. In interval notation: ( 5, ). Example 5: 1 Using interval notation, find the natural domain of x 3x+ Solution: x can be any number that makes the denominator non-zero. To find the exceptions, solve x 3x + = (x )(x 1) = x = ; x = 1. Thus x can be any number other than 1 or. Answer: The domain of f is (, 1) (1, ) (, ). How to find domain and range of function f from its graph The domain of f consists of all points on the horizontal axis that lie on vertical lines through points on the graph. The range of f consists of all points on the vertical axis that lie on horizontal lines through points on the graph. Example: Below is a graph given by formula F (x) = 5x. Keep on clicking to find the domain 3 and range of the function sketched. To find the domain, slide all points on the graph down to the x-axis. The domain of f is [1, ). To find the range, slide all points on the graph left to the y-axis. The range of f is [1, 6). ; (,6) (1,1) (1,) (,) Domain is [1, ) 1 x <
.1. Domain and range Example : Find the natural domain of the function 1 f(x) = x+5. Solution: The square root of a number less than is not defined. Thus we need x + 5, which means x 5. Division by zero is undefined. Thus we require x + 5 and so x 5. Answer: Combine the above requirements. The domain of f is x > 5. In interval notation: ( 5, ). Example 5: 1 Using interval notation, find the natural domain of x 3x+ Solution: x can be any number that makes the denominator non-zero. To find the exceptions, solve x 3x + = (x )(x 1) = x = ; x = 1. Thus x can be any number other than 1 or. Answer: The domain of f is (, 1) (1, ) (, ). How to find domain and range of function f from its graph The domain of f consists of all points on the horizontal axis that lie on vertical lines through points on the graph. The range of f consists of all points on the vertical axis that lie on horizontal lines through points on the graph. Example: Below is a graph given by formula F (x) = 5x. Keep on clicking to find the domain 3 and range of the function sketched. To find the domain, slide all points on the graph down to the x-axis. The domain of f is [1, ). To find the range, slide all points on the graph left to the y-axis. The range of f is [1, 6). ; (,6) (1,1) (1,) (,) Domain is [1, ) 1 x <
.1. Domain and range Example : Find the natural domain of the function 1 f(x) = x+5. Solution: The square root of a number less than is not defined. Thus we need x + 5, which means x 5. Division by zero is undefined. Thus we require x + 5 and so x 5. Answer: Combine the above requirements. The domain of f is x > 5. In interval notation: ( 5, ). Example 5: 1 Using interval notation, find the natural domain of x 3x+ Solution: x can be any number that makes the denominator non-zero. To find the exceptions, solve x 3x + = (x )(x 1) = x = ; x = 1. Thus x can be any number other than 1 or. Answer: The domain of f is (, 1) (1, ) (, ). How to find domain and range of function f from its graph The domain of f consists of all points on the horizontal axis that lie on vertical lines through points on the graph. The range of f consists of all points on the vertical axis that lie on horizontal lines through points on the graph. Example: Below is a graph given by formula F (x) = 5x. Keep on clicking to find the domain 3 and range of the function sketched. To find the domain, slide all points on the graph down to the x-axis. The domain of f is [1, ). To find the range, slide all points on the graph left to the y-axis. The range of f is [1, 6). ; (,6) Range is [1, 6) : 1 y < 6 (,6) (,1) (1,1) (1,) (,) Domain is [1, ) 1 x < All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 7
.1.5 Relations, Functions, and Graphs Please reread the discussion of the co-ordinate plane in Section 1.8. What is a relation between x and y? A graph is any set of points in the x,y-plane. A relation between x and y is any set of number pairs (x, y) with input x and output y. The graph of a relation is obtained by plotting all the number pairs of the relation as points in the x,y-plane One way to define a relation is by using an equation such as x + y =. The relation consists of precisely those number pairs (x, y) that satisfy the equation. Another description: the relation defined by an equation consists of all pairs (a, b) such that substituting a for x and b for y in the equation yields a true statement. For example, (, ) satisfies the equation x + y = : if you let x = and y =, you get the true statement () + () =. Thus (, ) is on the graph of the equation. However, (3, 1) does not satisfy the equation x + y =, since letting x = 3 and y = 1 yields a false statement 3 + (1) =. Therefore point (3, 1) is not on the graph of the equation. If you plot all points that do satisfy the equation, you will get a parabola. What is the graph of an equation? The graph of an equation in letters x and y consists of all points (x, y) in the x,y-plane that satisfy the equation. Example 6: Describe the graph of the equation x + y = as a set of points in the x,y-plane. Solution: First solve the equation for y = x. Answer: The ) graph of x + y = consists of all points (x, x where x is any real number. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 8
What is the graph of a function defined by f(x) := E? The graph of f is the graph of the equation y = f(x), and consists of all points (x, f(x)) in the x,y-plane where x is in the domain of f. Example 7: Describe the graph of each function. If f(x) = 3x +, the graph of f is is a slanted line passing through point (, ). If f(x) = 16 x, the graph of f is a parabola. If f(x) = sin(x), the graph of f is a wavy curve. How to recognize the graph of a function Vertical Line Test: A set of points in the x,y-plane is the graph of some function if and only if every vertical line meets the graph at one point or not at all. To see why, take any vertical line, for example x =. Then either is not in the domain of f, and so there is no point on the graph with input. Thus the line x = does not meet the graph. is in the domain of f. Then x is an input for f, with a single output f(). Thus there is only one point on that line, namely the point (, f()). Here s a bit of review of function definitions. How to substitute an expression in a function definition f(x) := E Substitute the expression for x. This means: Replace every x in E by the expression enclosed in parentheses. Example 8. Suppose h(x) = 16 x. Find and simplify h(a + b). Solution: Substitute a + b for x in the function formula. h(a + b) = 16 (a + b) = 16 [a + ab + b ] = 16 a ab b. In the following, you need to enclose every function value in parentheses before you simplify further. Exercise: Let f(x) = 3 x x, find and simplify a) f(x+h) f(x h) ; and b) f(3+h) f(3 h) h h Check your work by seeing whether substituting 3 for x in a) gives the answer to b). All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 9
.1.6 Examples of graphs of functions ou may know, and we will soon review, methods for graphing equations of several forms. Here are some examples. The graph of the equation y = 7 is the horizontal line through point (, 7). The graph of the equation x = 5 is the vertical line through point (5, ). The graph of the equation y = x + 7 is the straight line drawn through points (x, x + 7) obtained by choosing any two values for x. For example, choosing x = gives point (, 7), the y-intercept of the graph. Choose any other x-value, say x = 1, to find point (1, 11). The graph could be obtained by drawing the straight line through the points (, 7) and (1, 11). Exercise: Without using a graphing calculator, draw a rough sketch of the graph of the function y = x 1. Sometimes a function is defined in pieces. An important example is the absolute value function, which is written as abs(x) or as x. Here is the definition: If x, then x = x. If x, then x = x. { x if x This is also written as: x = x if x Example 9: Draw the graph of f(x) = x with domain [ 5, 5] Solution: 6 y = x y = f(x) = x y = x - - All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 1
.1.7 Why you must specify the domain of a function The three functions f(x) = x; g(x) = x ; and h(x) = x 3, are all drawn below with domain [ 1.5, 1.5]. The graphs are very different. However, if you restrict the domain to be the two x-values and 1, the graphs of all three functions are identical: the two black dots. h(x) = x 3 1.5 1.5 -.5-1 -1.5-1 -.5.5 1 g(x) = x f(x) = x Error warning: When you draw the graph of a function, choosing only whole numbers as inputs can produce a totally incorrect graph. For the simple function g(x) = x, choosing x =, 1,, 1,, connecting the points (x, x ) and smoothing out the curve gives a reasonable picture. But that s just a matter of luck. For more complicated functions, this method won t work. For example, suppose you use x =, 1,, 3, to sketch f(x) = 3x 5 3x + 15x 3 19x + 7x. As shown in the next frame, The tables and graphs of f(x) and g(x), using x =, 1,, 3, are identical. However, the graphs of f and g, using domain [, ] (all real numbers between and ), are very different. Exercise: Find f(), f(1), f(), f(3), and f() without using a calculator. A computer could plot 5 points instead of the 5 points that we are working with. Nevertheless, it could still make a serious mistake by connecting the dots to form the graph. Only THINKING about algebra (and calculus) allows you to be sure that the sketch you get is reasonably correct. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 11
16 1 8 At the left are the graphs of g(x) = x and f(x) = 3x 5 3x + 15x 3 19x + 7x with domains including the interval [, ]. The two functions have the same graph (5 black dots) for domain {, 1,, 3, } but very different graphs when the domain is the interval x. x f(x) g(x) 1 1 1 3 9 9 16 16 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 1
16 1 8 At the left are the graphs of g(x) = x and f(x) = 3x 5 3x + 15x 3 19x + 7x with domains including the interval [, ]. The two functions have the same graph (5 black dots) for domain {, 1,, 3, } but very different graphs when the domain is the interval x. x f(x) g(x) 1 1 1 3 9 9 16 16 x f(x) g(x).5.5 1.9375 1.5.5-1.96875.5 6.5 1.6875 3.5 1.5.65.5.5 18.8375 Example 1: Multiply out and rewrite x + 3x(x 1)(x )(x 3)(x ) as a polynomial. Answer: 3x 5 3x + 15x 3 19x + 7x, which is f(x). Exercise: Prove that f(x) = g(x) = x for x =, 1,, 3, but for no other values of x 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 1
.1.8 Section.1 Quiz Review Example 1: Suppose the height of a ball above the ground after t seconds is h(t) = 1 16t. How high above the ground is the ball after seconds? Example : When does the ball hit the ground? Example 3: Find the natural domain of the function f(x) = Example : Using interval notation, find the natural domain of 1 x+5. Example 5: Draw the graph of y = f(x) = 5x 3 for 1 x <. Then find the domain and range of the function f. 1 x 3x+ Example 6: Describe the graph of the equation x + y = as a set of ordered pairs. Example 7: Describe the graph of the following functions. f(x) = 3x + g(x) = 16 x Example 8: Suppose h(x) = 16 x. Find and simplify h(a + b). Example 9: Draw the graph of f(x) = x with domain [ 5, 5] Example 1: Multiply out and rewrite x + 3x(x 1)(x )(x 3)(x ) as a polynomial. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.1: What is a function? 8/9/16 Frame 13
..1: Graphing a real-life function..: The effect of the choice of domain..3: Special features of graphs..: Linear functions defined on intervals..5: Graphing a polynomial function on an interval..6: Piecewise defined functions..7: A three-part piecewise linear graph..8: The vertical line test..9: Quiz review All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 1
..1 Graphing a real-life function If a function describes a real-life process, you can understand that process better by studying the function s graph. Here s an example from physics. Throw a ball up from ground level and release it with velocity 6 feet per second (how many miles per hour is that?). After t seconds, the height of the ball, measured in feet above ground level, is y = h(t) = 6t 16t. In this example, the function s input is time and its output is height. The ordered number pair (input,output) is written in English as (time,height) and in math as (t, h(t)). In the real-life problem described by this function, the ball is released at time t = The function s domain starts at t =. and ends when the ball hits the ground. To find when that happens, set the height function h(t) = 6t 16t equal to and solve 6t 16t = 16t( t) = t =, t =. Therefore the domain of h is t. To get some idea of how the ball s height changes, make a table of values for t, with spacing.5 between t-values. Then plot the pairs in the right column of the table. t h(t) = 6t 16t (t, h(t)). = (, ).5 3 = 8 (.5, 8) 1. 6 16 = 8 (1, 8) 1.5 96 36 = 6 (1.5, 6). 18 6 = 6 (, 6).5 16.5 1.5 = 6 (.5, 6) 3. 16 3 1 = 8 (3, 8) 3.5 16 3.5.5 = 8 (3.5, 8). 16 = (, ) 7 6 5 3 1.5 1 1.5.5 3 3.5 How should we connect the dots? It appears that the ball reaches its maximum height of 6 feet after seconds. How can we be sure? All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 15
We can connect dots with line segments by hand 7 6 5 3 1.5 1 1.5.5 3 3.5 or we can plot more points (using a computer) to avoid sharp corners. Looking at these tables and graphs suggests that the ball s maximum height is 6 feet, when t = seconds. However, appearances can be deceiving. Exercise: Prove that the maximum value of h(t) = 6 t is h() = 6. Hint: Start with the fact that t for all real numbers t. A large part of first semester calculus is devoted to understanding and sketching graphs of complicated functions. The most important features are called local maximum and minimum points of the function. These are hilltops or valley bottoms that appear on the graph of the function. 7 6 5 3 1.5 1 1.5.5 3 3.5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 16
.. The shape of a graph depends on the domain you choose. Suppose you are asked to graph y = f(x) = x. ou know that the graph should look like a bowl. ou need to tell the computer a domain of x-values and a range of y-values. ou try some possibilities as shown below..1 1.5 -.5 -.1 -.1 -.5.5.1 5-5 -1-1 -5 5 1 Neither graph above gives what you expect. To find the interesting features of a graph, you need to choose the viewing window intelligently. The techniques of first semester calculus will show you how to do that. Obviously you would prefer the viewing windows at the right to the ones above. 1.5 -.5-1 -1 -.5.5 1 5.5 -.5-5 -5 -.5.5 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 17
..3 Special features of graphs Exercise. Click on Wolfram Grapher. Type in the following: Plot y = xˆ for 1 <= x <= 1 and 1 <= y <= 1 Then hit Enter to see the graph. Revise the command you typed to find a good viewing window for each of the following functions. 1. f(x) = x. g(x) = x 3. h(x) = x 3 On each graph, try to locate the following features:. p(x) = 5x x 1 5. q(x) = x 3 3x + 9x 6. r(x) = 1 x 9x hilltop valley bottom rising plateau falling plateau In Section 3.1, we will show how to use algebra to discover the one interesting feature on the graph of a quadratic function. The method involves completing the square. For more complicated functions, such as numbers 3,5, and 6 above, calculus is required. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 18
.. Drawing a graph of a linear function whose domain is an interval Finite intervals from a to b a x b is the closed interval [a, b]. a < x < b is the open interval (a, b). a x < b is the half-open interval [a, b). a < x b is the half-closed interval (a, b]. How to graph f(x) = mx + b on a finite interval For each interval endpoint x, plot point (x, f(x)) : as a filled dot if the endpoint is in the interval; or as a hollow dot if the endpoint is not in the interval. Connect the endpoints with a line segment. Example 1: Draw the graph of y = f(x) = x + 1 for domain 3 < x 1. Solution: Click through the steps: 3 1-1 - -3 - - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 19
.. Drawing a graph of a linear function whose domain is an interval Finite intervals from a to b a x b is the closed interval [a, b]. a < x < b is the open interval (a, b). a x < b is the half-open interval [a, b). a < x b is the half-closed interval (a, b]. How to graph f(x) = mx + b on a finite interval For each interval endpoint x, plot point (x, f(x)) : as a filled dot if the endpoint is in the interval; or as a hollow dot if the endpoint is not in the interval. Connect the endpoints with a line segment. 3 1-1 Example 1: Draw the graph of y = f(x) = x + 1 for domain 3 < x 1. Solution: Click through the steps: Since left endpoint x = 3 is not in the interval, plot point ( 3, f( 3)) = P ( 3, ) as a hollow dot. - -3 P(-3,-) - - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 19
.. Drawing a graph of a linear function whose domain is an interval Finite intervals from a to b a x b is the closed interval [a, b]. a < x < b is the open interval (a, b). a x < b is the half-open interval [a, b). a < x b is the half-closed interval (a, b]. 3 Q(1,) How to graph f(x) = mx + b on a finite interval For each interval endpoint x, plot point (x, f(x)) : as a filled dot if the endpoint is in the interval; or as a hollow dot if the endpoint is not in the interval. Connect the endpoints with a line segment. 1-1 Example 1: Draw the graph of y = f(x) = x + 1 for domain 3 < x 1. Solution: Click through the steps: Since left endpoint x = 3 is not in the interval, plot point ( 3, f( 3)) = P ( 3, ) as a hollow dot. Since right endpoint x = 1 is in the interval, plot point (1, f(1)) = Q(1, ) as a solid dot. - -3 P(-3,-) - - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 19
.. Drawing a graph of a linear function whose domain is an interval Finite intervals from a to b a x b is the closed interval [a, b]. a < x < b is the open interval (a, b). a x < b is the half-open interval [a, b). a < x b is the half-closed interval (a, b]. 3 Q(1,) How to graph f(x) = mx + b on a finite interval For each interval endpoint x, plot point (x, f(x)) : as a filled dot if the endpoint is in the interval; or as a hollow dot if the endpoint is not in the interval. Connect the endpoints with a line segment. 1-1 Example 1: Draw the graph of y = f(x) = x + 1 for domain 3 < x 1. Solution: Click through the steps: Since left endpoint x = 3 is not in the interval, plot point ( 3, f( 3)) = P ( 3, ) as a hollow dot. Since right endpoint x = 1 is in the interval, plot point (1, f(1)) = Q(1, ) as a solid dot. Draw the line segment from P to Q. - -3 P(-3,-) - - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 19
..5 Drawing a graph of a polynomial function whose domain is an interval Example : Draw the graph of y = f(x) = x for domain 3 < x 1. Solution:We know that the graph of y = x is a parabola. We want to draw exactly those points on the graph with x-coordinate satisfying 3 < x 1. 9 8 7 6 5 3 1 Solution: - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..5 Drawing a graph of a polynomial function whose domain is an interval Example : Draw the graph of y = f(x) = x for domain 3 < x 1. Solution:We know that the graph of y = x is a parabola. We want to draw exactly those points on the graph with x-coordinate satisfying 3 < x 1. 9 8 7 6 P(-3,9) Solution: When x = 3, y = f(x) = x = 9. The left endpoint of the graph (shown as a hole since x = 3 is missing from the domain) is the point P ( 3, 9). 5 3 1 - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..5 Drawing a graph of a polynomial function whose domain is an interval Example : Draw the graph of y = f(x) = x for domain 3 < x 1. Solution:We know that the graph of y = x is a parabola. We want to draw exactly those points on the graph with x-coordinate satisfying 3 < x 1. 9 8 7 6 5 P(-3,9) Solution: When x = 3, y = f(x) = x = 9. The left endpoint of the graph (shown as a hole since x = 3 is missing from the domain) is the point P ( 3, 9). When x = 1, y = f(x) = x = 1. The right endpoint of the graph (shown as a filled dot since x = 1 is included in the domain) is the point Q(1, 1). 3 1 Q(1,1) - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..5 Drawing a graph of a polynomial function whose domain is an interval Example : Draw the graph of y = f(x) = x for domain 3 < x 1. Solution:We know that the graph of y = x is a parabola. We want to draw exactly those points on the graph with x-coordinate satisfying 3 < x 1. 9 8 7 6 5 3 P(-3,9) Solution: When x = 3, y = f(x) = x = 9. The left endpoint of the graph (shown as a hole since x = 3 is missing from the domain) is the point P ( 3, 9). When x = 1, y = f(x) = x = 1. The right endpoint of the graph (shown as a filled dot since x = 1 is included in the domain) is the point Q(1, 1). To finish the graph, fill in some other points (say (, ) and ( 1, 1)). 1 Q(1,1) - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..5 Drawing a graph of a polynomial function whose domain is an interval Example : Draw the graph of y = f(x) = x for domain 3 < x 1. Solution:We know that the graph of y = x is a parabola. We want to draw exactly those points on the graph with x-coordinate satisfying 3 < x 1. 9 8 7 6 5 3 1 P(-3,9) Q(1,1) Solution: When x = 3, y = f(x) = x = 9. The left endpoint of the graph (shown as a hole since x = 3 is missing from the domain) is the point P ( 3, 9). When x = 1, y = f(x) = x = 1. The right endpoint of the graph (shown as a filled dot since x = 1 is included in the domain) is the point Q(1, 1). To finish the graph, fill in some other points (say (, ) and ( 1, 1)). Draw a smooth curve, which you should recognize as the graph of the parabola y = x with domain 3 < x 1. - -3 - -1 1 3 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..6 Piecewise defined functions Sometimes a function is defined piecewise, using different formulas on different intervals. To draw the graph of such a function, just follow the procedure in the previous frame to plot the graph on one interval at a time. { x + 1 if 3 < x < 1 Example 3: draw the graph of f(x) = x if 1 x < 3 1-1 - -3 - -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 1
..6 Piecewise defined functions Sometimes a function is defined piecewise, using different formulas on different intervals. To draw the graph of such a function, just follow the procedure in the previous frame to plot the graph on one interval at a time. { x + 1 if 3 < x < 1 Example 3: draw the graph of f(x) = x if 1 x < 3 1-1 Q(1,) Solution: f(x) = x + 1 for 3 < x < 1 is a line segment from (but omitting) ( 3, f(3)) = P ( 3, ) to (but omitting) point (1, f(1)) = Q(1, ), as shown at the left. - P(-3,-) -3 - -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 1
..6 Piecewise defined functions Sometimes a function is defined piecewise, using different formulas on different intervals. To draw the graph of such a function, just follow the procedure in the previous frame to plot the graph on one interval at a time. { x + 1 if 3 < x < 1 Example 3: draw the graph of f(x) = x if 1 x < 3 1-1 - -3 P(-3,-) Q(1,) R(1,1) - -5 - -3 - -1 1 3 5 Q(,-) Solution: f(x) = x + 1 for 3 < x < 1 is a line segment from (but omitting) ( 3, f(3)) = P ( 3, ) to (but omitting) point (1, f(1)) = Q(1, ), as shown at the left. f(x) = x for 1 x < is a line segment from (and including) (1, f(1)) = R(1, 1) to (but omitting ) point (, f()) = S(, ). This completes the graph. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 1
Sometimes you want to draw the graph of the part of a line with domain an infinite interval such as (, ) (same as x < ) or [, ) (same as x). Example : Draw the graph of y = x + 1 for x <. 3 1-1 - -3 - -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
Sometimes you want to draw the graph of the part of a line with domain an infinite interval such as (, ) (same as x < ) or [, ) (same as x). Example : Draw the graph of y = x + 1 for x <. 3 1 P(,3) Solution: y = x + 1 for x < is a line segment to the left of (but not including) point (, f()) = P (, 3). Plot point P as a hollow dot. -1 - -3 - -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
Sometimes you want to draw the graph of the part of a line with domain an infinite interval such as (, ) (same as x < ) or [, ) (same as x). Example : Draw the graph of y = x + 1 for x <. 3 1-1 - -3 Q(,1) P(,3) - -5 - -3 - -1 1 3 5 Solution: y = x + 1 for x < is a line segment to the left of (but not including) point (, f()) = P (, 3). Plot point P as a hollow dot. ou need to plot any other point to the left of P, with x-coordinate less than. For example, choose x = to obtain point Q(, 1). Plot this point as a filled dot: it is on the graph. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
Sometimes you want to draw the graph of the part of a line with domain an infinite interval such as (, ) (same as x < ) or [, ) (same as x). Example : Draw the graph of y = x + 1 for x <. 3 1-1 - -3 Q(,1) P(,3) - -5 - -3 - -1 1 3 5 Solution: y = x + 1 for x < is a line segment to the left of (but not including) point (, f()) = P (, 3). Plot point P as a hollow dot. ou need to plot any other point to the left of P, with x-coordinate less than. For example, choose x = to obtain point Q(, 1). Plot this point as a filled dot: it is on the graph. Draw the line from P to Q and then to an arrowhead near the left of the grid. The arrowhead is used to indicate that the line continues (forever) toward the left. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..7 A three-part piecewise linear graph 3x if x Example 5: Draw the graph of f(x) = x + 1 if < x 1 7 if 1 < x 9 6 3 Solution: -3-6 -9-1 -15-5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 3
..7 A three-part piecewise linear graph 3x if x Example 5: Draw the graph of f(x) = x + 1 if < x 1 7 if 1 < x 9 6 3-3 -6 P(-,-6) Solution: y = 3x for x starts at (and includes) P (, 6). ou need another point on the line. Choose x = 3 to get point Q( 3, 9). Draw the line from P to Q to an arrowhead at the left edge of the grid. -9 Q(-3,-9) -1-15 -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 3
..7 A three-part piecewise linear graph 3x if x Example 5: Draw the graph of f(x) = x + 1 if < x 1 7 if 1 < x 9 6 3-3 -6 R(-,-1) P(-,-6) S(1,) Solution: y = 3x for x starts at (and includes) P (, 6). ou need another point on the line. Choose x = 3 to get point Q( 3, 9). Draw the line from P to Q to an arrowhead at the left edge of the grid. y = x + 1 for < x 1 goes from (but omits) point R(, 1) to (and includes) point S(1, ). -9 Q(-3,-9) -1-15 -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 3
..7 A three-part piecewise linear graph 3x if x Example 5: Draw the graph of f(x) = x + 1 if < x 1 7 if 1 < x 9 6 3-3 -6-9 -1 R(-,-1) P(-,-6) Q(-3,-9) T(1,7) S(1,) U(3,7) -15-5 - -3 - -1 1 3 5 Solution: y = 3x for x starts at (and includes) P (, 6). ou need another point on the line. Choose x = 3 to get point Q( 3, 9). Draw the line from P to Q to an arrowhead at the left edge of the grid. y = x + 1 for < x 1 goes from (but omits) point R(, 1) to (and includes) point S(1, ). y = 7 for 1 < x starts at (but omits) point T (1, 7). ou need another point on the line. Choose x = 3 to get point U(3, 7). Draw the line from T to U to an arrowhead at the right edge of the grid. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 3
..8 Checking that a graph is the graph of a function: the vertical line test Vertical line test A graph is the graph of a function provided every vertical line meets the graph once or never. { x + 1 if 3 < x < 1 Below we redraw the graph of f(x) = from Example 3. x if 1 x < Example 6: 3 1 Use the vertical line test to decide if the graph below is the graph of a function. Q(1,) R(1,1) Solution: Point R(1, 1) is on the graph, but Q(1, ) is not. Therefore the vertical blue line x = 1 meets the graph at the single point Q(1, ). -1 - -3 P(-3,-) Q(,-) - -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..8 Checking that a graph is the graph of a function: the vertical line test Vertical line test A graph is the graph of a function provided every vertical line meets the graph once or never. { x + 1 if 3 < x < 1 Below we redraw the graph of f(x) = from Example 3. x if 1 x < Example 6: 3 1-1 Use the vertical line test to decide if the graph below is the graph of a function. Q(1,) R(1,1) Solution: Point R(1, 1) is on the graph, but Q(1, ) is not. Therefore the vertical blue line x = 1 meets the graph at the single point Q(1, ). The vertical line through point (x, ) meets the graph exactly once for 3 < x <, which is the domain of f. - -3 P(-3,-) Q(,-) - -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..8 Checking that a graph is the graph of a function: the vertical line test Vertical line test A graph is the graph of a function provided every vertical line meets the graph once or never. { x + 1 if 3 < x < 1 Below we redraw the graph of f(x) = from Example 3. x if 1 x < Example 6: 3 1-1 - -3 Use the vertical line test to decide if the graph below is the graph of a function. P(-3,-) Q(1,) R(1,1) - -5 - -3 - -1 1 3 5 Q(,-) Solution: Point R(1, 1) is on the graph, but Q(1, ) is not. Therefore the vertical blue line x = 1 meets the graph at the single point Q(1, ). The vertical line through point (x, ) meets the graph exactly once for 3 < x <, which is the domain of f. For all other x-values, the vertical line through point (x, ) does not meet the graph. Answer: Every vertical line meets the graph once or never. Therefore the graph passes the vertical line test and is the graph of a function. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame
..9 Chapter. Quiz Review Example 1: Draw the graph of y = f(x) = x + 1 for domain 3 < x 1. Example : Draw the graph of y = f(x) = x for domain 3 < x 1. { x + 1 if 3 < x < 1 Example 3: Draw the graph of f(x) = x if 1 x < Example : Draw the graph of y = x + 1 for x <. 3x if x Example 5: Draw the graph of f(x) = x + 1 if < x 1 7 if 1 < x Example 6: Use the vertical line test to decide if the graph below is the graph of a function. 3 1 Q(1,) R(1,1) -1 - -3 P(-3,-) Q(,-) - -5 - -3 - -1 1 3 5 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.: Graphs of functions 8/9/16 Frame 5
Section.3: Analyzing graphs of functions.3.1: Maximum and minimum points.3.: Function increasing or decreasing.3.3: Given h(t), find t.3.: The graph of a degree 3 polynomial.3.5: How many solutions of h(t) = K?.3.6: Quiz review All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.3: Analyzing graphs of functions 8/9/16 Frame 6
Getting basic information from the graph of a function A function is a collection of number pairs (x, y), where x is an input and y = f(x) is the corresponding output. Here are some models that give rise to functions. Manufacturing: Input x is how many tons of bricks a factory sells. Output f(x) is net profit: the sale price of x bricks minus their cost. Saving the dolphins: Input t is the number of hours after Noon. Output f(t) is the gap between the radius of a circular oil slick spreading from an oil tanker explosion and a dolphin swimming away from the explosion, as in Section 1.6. Moving particle: Input t is the number of seconds after Noon. Output h(t) is the height above ground at time t of a ball thrown up from ground at Noon. Each of these physical situations is modeled by a different function. However, the information we would like to know about each model can usually be obtained by asking similar questions about the functions s graph. Suppose you know the output y = h(t). What input or inputs t produced that output? What is the maximum value of y = h(t)? What is the minimum value? In what (time) interval of inputs is y = h(t) getting larger as t increases? In what (time) interval is y = h(t) getting smaller as t increases? A ball is thrown up at time t = and released with velocity 96 feet per second. Its height above the ground (until it comes back down) is y = h(t) = 96t 16t feet. Below is the graph of h with domain t 6. y 16 18 96 6 3 1 3 5 6 All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.3: Analyzing graphs of functions 8/9/16 Frame 7 t
.3.1 Maximum and minimum points Maximum and minimum points of a function graph Suppose f is a function defined on an interval I and a is a point in I. Then point (a, f(a)) on the graph of f is an absolute maximum point means: f(a) f(x) for all x in I. an absolute maximum point means: f(a) f(x) for all x in I. a relative maximum point (the top of a hill) means: f(a) f(x) for all x in some small interval (a d, a + d) contained in I. a relative minimum point (the bottom of a valley) means: f(a) f(x) for all x in some small interval (a d, a + d) contained in I. Example 1: Find all absolute or relative maximum or minimum points on the graph below, with domain x 6. y 16 18 96 6 3 A function can t have a relative maximum or minimum point at an endpoint of a closed interval I, since I doesn t contain any interval around either endpoint. Maximum and minimum values of a function f f(a) is a maximum (minimum) value of function f means: (a, f(a)) is a maximum (minimum) point of the graph of f. 1 3 5 6 Answer: t All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.3: Analyzing graphs of functions 8/9/16 Frame 8
.3.1 Maximum and minimum points Maximum and minimum points of a function graph Suppose f is a function defined on an interval I and a is a point in I. Then point (a, f(a)) on the graph of f is an absolute maximum point means: f(a) f(x) for all x in I. an absolute maximum point means: f(a) f(x) for all x in I. a relative maximum point (the top of a hill) means: f(a) f(x) for all x in some small interval (a d, a + d) contained in I. a relative minimum point (the bottom of a valley) means: f(a) f(x) for all x in some small interval (a d, a + d) contained in I. Example 1: Find all absolute or relative maximum or minimum points on the graph below, with domain x 6. y 16 Absolute and relative Maximum 18 96 6 3 P(3,1) A function can t have a relative maximum or minimum point at an endpoint of a closed interval I, since I doesn t contain any interval around either endpoint. Maximum and minimum values of a function f f(a) is a maximum (minimum) value of function f means: (a, f(a)) is a maximum (minimum) point of the graph of f. 1 3 5 6 Answer: (3, 1) is both an absolute maximum point and a relative maximum point. t All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.3: Analyzing graphs of functions 8/9/16 Frame 8
.3.1 Maximum and minimum points Maximum and minimum points of a function graph Suppose f is a function defined on an interval I and a is a point in I. Then point (a, f(a)) on the graph of f is an absolute maximum point means: f(a) f(x) for all x in I. an absolute maximum point means: f(a) f(x) for all x in I. a relative maximum point (the top of a hill) means: f(a) f(x) for all x in some small interval (a d, a + d) contained in I. a relative minimum point (the bottom of a valley) means: f(a) f(x) for all x in some small interval (a d, a + d) contained in I. Example 1: Find all absolute or relative maximum or minimum points on the graph below, with domain x 6. y 16 Absolute and relative Maximum 18 96 6 3 P(3,1) A function can t have a relative maximum or minimum point at an endpoint of a closed interval I, since I doesn t contain any interval around either endpoint. Maximum and minimum values of a function f f(a) is a maximum (minimum) value of function f means: (a, f(a)) is a maximum (minimum) point of the graph of f. Q(,) Absolute Min Absolute Min R(6,) t 1 3 5 6 Answer: (3, 1) is both an absolute maximum point and a relative maximum point. (, ) and (6, ) are absolute minimum points, but they are not relative minima, since they are at the endpoints of the closed interval [, 6]. All rights reserved. Copyright 16 by Stanley Ocken CCN Math Review Section.3: Analyzing graphs of functions 8/9/16 Frame 8