A relation is a set of ordered pairs of real numbers. The domain, D, of a relation is the set of all first coordinates of the ordered pairs in the relation (the xs). The range, R, of a relation is the set of all second coordinates of the ordered pairs in the relation (the ys). In graphing relations, the horizontal axis is called the domain axis and the vertical axis is called the range axis. The domain and range of a relation can often be determined from the graph of the relation. **If the domain or range consists of a finite number of points, use braces and set notation. **If the domain or range consists of intervals of real numbers, use interval (or inequality) notation. State the domain and range of the relation. EX: {(-1,1), (1,5), (0,3)} 1 P a g e
A function is a special kind of relation that pairs each element of the domain with one and only one element of the range. (For every x there is exactly one y.) A function is a correspondence between a first set, domain, and a second set, range. In a function no two ordered pairs have the same first coordinate. That is, each first coordinate appears only once. Although every function is by definition a relation, not every relation is a function. EX: Which of the following relations are functions? ( 2, 8),(3, 0),( 1, 5), ( 2, 5),(3, 5),( 1, 5), (2, 5),(3, 0),(2, 0) To determine whether or not the graph of a relation represents a function, we apply the vertical line test which states that if any vertical line intersects the graph of a relation in more than one point, then the relation graphed is not a function. 2 P a g e
Is the relation a function? EX: 3 P a g e
Function notation and evaluating functions (finding range values)... EXAMPLE: ( x) 2x 3 to 2x 3. f is the name of the function. f is read f of x is equal x is representative of an element in the domain of f. f (x) is representative of an element of the range of f, and means the same as y. 2x 3 is the function rule. EVALUATE: f ( 5) f ($) f (x 1) 4 P a g e
Examples: Graph and find the domain and range of the following functions f(x) = x 2 5 f(x) = x 3 x 5 P a g e
Find the domain and range of the following f(x) = Determine whether given points f(1) and f(3) are in the domain of f(x) = Restricted values occur when the denominator is equal to zero. (Why? because a zero in the denominator makes a function undefined). Also restricted values occur when the value under the radical is less than zero (meaning negative) for even indexed roots. 6 P a g e
a. g(x) = b. h(x) = c. ( ) 7 P a g e
To graph functions using a graphing calculator. Step 1: Hit the y= button (purple) located under the screen on the left Step2: You will see y1= y2= You can enter your equation now For instance if we wanted to graph y=2x+3 then you would enter 2x+3 on this screen. Step 3: Hit enter Step 4: Hit the Graph button (purple) located under the screen on the right. This step will graph the function for you. Note if you cannot see your graph then your window settings are not set correctly. You need to hit the window button (purple) located under the window. You should have the x and y max be 10 and the x and y min be -10, the increment should be 1. You can also evaluate function values after you have entered your function into the y1=. Let s say your function is f(x) = 2x+3 and you have this saved in y1= then you can determine f(30) by simply choosing y1 hit enter open parenthesis then 30 then close parenthesis then enter and your calculator will calculate this for you. the answer it will give you is 63. 8 P a g e
Review of the rectangular coordinate system axes, origin, quadrants, ordered pairs, coordinates, signs of coordinates of points, etc. Y-Axis Vertical Axis Quadrant II Negative x-values Positive y-values ( -, + ) Quadrant I Positive x-values Positive y-values ( +, + ) X-Axis Horizontal Axis Quadrant III Negative x-values Negative y-values ( -, - ) Quadrant IV Positive x-values Negative y-values ( +, - ) Plot means to show the location of a point on the rectangular coordinate system. Ordered Pair: ( x, y ) x: is the x-coordinate (move on the x-axis) 1 st coordinate y: is the y-coordinate (move on the y-axis) 2 nd coordinate 9 P a g e
Axis Title Obtaining Information from Graphs You can obtain information about a function from its graph. At the right or left of a graph you will find closed dots, open dots or arrows. - Closed dots mean that the graph does not extend beyond this point, and the point belongs to the graph - Open dots mean that the graph does not extend beyond this point, and the point does not belong to the graph - An arrow indicates that the graph extends indefinitely in the direction in which the arrow points. Example- 5 Series 1 4 3 2 1 0 Category 1 Category 2 Category 3 Category 4 Using the above graph give the following: a) Explain why represents the graph of a function b) Use the graph to determine what is ( ) c) For what categories of ( ) is the output value greater than 3.5. 10 P a g e
Identifying Intercepts Two distinct points determine a line. The points on a line have coordinates that make the equation of the line true. To find the y-intercept of a line, let x=0. To find the x-intercept of a line, let y=0. Problem Type #1: Given the equation of a line, you should be able to find the intercepts. Graph the lines too. EX 1: 11 P a g e
EX 2. 12 P a g e
If the graph of a function rises from left to right, it is said to be increasing. If the graph of a function falls from left to right, it is said to be decreasing. If the function values stay the same from left to right, it is said to be constant. Increasing Decreasing Constant For the following figures determine the intervals for which the function is increasing, decreasing, or constant. 13 P a g e
Relative Maximum and Minima c 1 c 2 c 3 For the graph above note the peaks and valleys at the x-values c 1, c 2, c 3. The function value f(c 2 ) is called the relative maximum, f(c 1 ) and f(c 3 ) are called relative minimum. Simply stated, f(c) is a relative maximum if f(c) is the highest point in some open interval, and f(c) is a relative minimum if f(c) is the lowest point in some open interval. Example: For the graph below determine the relative maxima and minima of the function ( ) and the intervals on which the function is increasing or decreasing. 14 P a g e
Symmetry Symmetric with Respect to the y-axis: if for every point (x,y) on the graph, the point (-x,y) is also on the graph. (-x, y) (x, y) Symmetric with respect to the x-axis: if for every point (x,y) on the graph, the point (x,-y) is also on the graph. Symmetric with respect to the origin: if for every point (x,y) on the graph, the point (-x,-y) is also on the graph. (-x,- y) (x, y) (x,- y) (x, y) For the given graphs determine the symmetry. 15 P a g e
Even and Odd Functions- Even If the graph of a function f is symmetric with respect to the y-axis. For each x in the domain of f, ( ) ( ) Odd If the graph of a function f is symmetric with respect to the origin. For each x in the domain of f, ( ) ( ) Procedure for Determining Even and Odd 1. Find ( ) and simplify. If ( ) ( ) then f is even 2. Find ( ) and simplify and compare with ( ). If ( ) ( ) then f is odd Example Determine Even, Odd or Neither a) ( ) b) ( ) 16 P a g e
Piecewise Functions- A function that is determined by two or more equations over a specific domain. Example Graph the piecewise functions a) ( ) { 1. Give the domain and range of g(x) 2. Find ( ) 17 P a g e
b) ( ) { 1. Give the domain and range of f(x) 2. Find ( ) 18 P a g e
Difference Quotient f(x+h) f(x) Secant Line (x, f(x)) (x+h, f(x+h)) f Looking at the nonlinear function f if you draw a line through two points (x, f(x)) and (x+h, f(x+h)). The slope of that line, called the secant line is the equation of the difference quocient. x x+h This is the difference quotient or average rate of change. ( ) ( ) ( ) ( ) ( ) Example For the function f given by ( ) difference quotient., find the 19 P a g e
Example For the function f given by ( ) difference quotient., find the 20 P a g e