3.1 Functions. The relation {(2, 7), (3, 8), (3, 9), (4, 10)} is not a function because, when x is 3, y can equal 8 or 9.

Transcription

1 3. Functions Cubic packages with edge lengths of cm, 7 cm, and 8 cm have volumes of 3 or cm 3, 7 3 or 33 cm 3, and 8 3 or 5 cm 3. These values can be written as a relation, which is a set of ordered pairs, (, ). The relation is {(, ), (7, 33), (8, 5)}. An set of ordered pairs is a relation. A function is a special tpe of relation. A function is a set of ordered pairs in which, for ever value of, there is onl one value of. The relation {(, 5), (, ), (3, 7), (, 8)} is a function because, for each value of, there is onl one value of. 8 The relation {(, 7), (3, 8), (3, 9), (, )} is not a function because, when is 3, can equal 8 or 9. 8 The following graphs model two relations. Relation A Relation B 7 MHR Chapter 3

2 Relation A is a function. For each value of, there is onl one value of. For eample, when is 3, is 3. Relation B is not a function. For each value of, ecept, there are two values of. For eample, when is, is or. INVESTIGATE & INQUIRE The packaging industr is an important application of mathematics. Packages are made in man shapes and sizes. An open-topped bo can be made from a square sheet of tin b cutting out smaller squares from the corners and folding up the sides. The diagrams show that removing squares of side length cm from the corners of an 8 cm b 8 cm sheet of tin results in an open-topped bo with dimensions cm b cm b cm and volume or 5 cm 3. cm cm 8 cm cm cm cm 8 cm. Cop and complete a table, like the one shown, or set up a spreadsheet to determine the following. Side Length of Removed Squares (cm) 3 Dimensions of Bo (cm) Volume of Bo (cm 3 ) a) the maimum volume of an open-topped bo that can be made from an 8 cm b 8 cm piece of tin b removing smaller squares with side lengths that are whole numbers of centimetres b) the side length of the smaller squares that result in the maimum volume of the bo 5 3. Functions MHR 7

3 . a) From question, list a set of ordered pairs in the form (side length of removed squares, volume of bo). b) Graph the volume of the bo versus the side length of the removed squares. c) Does the graph model a function? Eplain. V Volume Side length s 3. a) Graph the side length of the removed squares versus the volume of the bo. b) Does the graph model a function? Eplain.. Eplain wh a packaging compan might be interested in the answers to question. s Side length Volume V One wa to determine if a relation is a function is to graph the relation and then use the vertical line test. If an vertical line passes through more than one point on the graph, then the relation is not a function. EXAMPLE Vertical Line Test Determine if each relation is a function. a) b) SOLUTION a) The relation is a function. No vertical line passes through more than one point. 7 MHR Chapter 3

4 b) The relation is not a function. The vertical line shown passes through two points, (, ) and (, ), so there are two values of for the same value of. EXAMPLE Falling Object For an object falling under the effect of gravit, the approimate distance fallen, d metres, is given b the equation d = 5t, where t seconds is the time since the object was dropped. a) Graph the equation. b) Determine if the relation is a function. SOLUTION a) The distance, d, depends on the time, t, so we call d the dependent variable and t, the independent variable. It is customar to graph the dependent variable versus the independent variable, that is, with the dependent variable on the vertical ais and with the independent variable on the horizontal ais. Graph d versus t using paper and pencil, a graphing calculator, or graphing software. Note that the graph lies in the first quadrant because the values of d and t cannot be negative. t 3 5 d d Distance (m) 8 d = 5t For this graph, the window variables include Xmin =, Xma =, Ymin =, and Yma = Time (s) b) Because, for ever value of t, there is onl one value of d, the relation is a function. t 3. Functions MHR 73

5 Note that, for a function, the dependent variable is said to be a function of the independent variable. In Eample, the distance, d, depends on, and is a function of, the time, t. The set of the first elements in a relation is called the domain of the relation. For the relation {(, ), (3, ), (5, ), (7, 8)}, the domain is {, 3, 5, 7}. The set of the second elements in a relation is called the range of the relation. For the relation {(, ), (3, ), (5, ), (7, 8)}, the range is {,,, 8}. A function can be defined as a set of ordered pairs in which, for each element in the domain, there is eactl one element in the range. EXAMPLE 3 Determining the Domain and Range State the domain and range of each relation. Determine if each relation is a function. a) {(, ), ( 3, 3), (, ), ( 3, 5), (, )} b) = c) = + 3 SOLUTION a) The domain is {, 3,, }. The range is {, 3,, 5, }. The relation is not a function because, for the element 3 in the domain, there are two elements, 3 and 5, in the range. b) Graph the relation = using paper and pencil, a graphing calculator, or graphing software For this graph, the window variables include Xmin = 5, Xma = 5, Ymin =, and Yma = 9. = MHR Chapter 3

6 Since can be an real number, the domain is the set of real numbers. Since the value of is alwas greater than or equal to zero, the range is. The relation is a function because, for each element in the domain, there is eactl one element in the range. c) Graph the relation = + 3 using paper and pencil, a graphing calculator, or graphing software = + 3 For this graph, the window variables include Xmin = 3, Xma = 5, Ymin =, and Yma = 8. Since can be an real number, the domain is the set of real numbers. Since can be an real number, the range is the set of real numbers. The relation is a function because, for each element in the domain, there is eactl one element in the range. In an equation such as = + 3, depends on and is a function of. An equation that is a function can be named using function notation. - notation function notation = + 3 f() = + 3 In function notation, f names a function. Notice that the smbol f() is another name for. The smbol f() is the value of the function f at. Read f() as the value of f at or f of. A function is like a machine. When an -value in the domain of the function f enters, the machine produces the output f(). The output f() is determined b the rule of the function. input f f() The machine produces eactl one output, f(), for each -value. 3. Functions MHR 75

7 To find f(5) for the function f() = + 3, substitute 5 for in f() = + 3. When = 5, the value of, or f(5), is 3, because = 3. f(5) = 3 8 f() = + 3 = 5 EXAMPLE Evaluating a Function If f() = 3 +, find a) f() b) f( ) c) f() SOLUTION a) f() = 3 + f() = 3() + = 9 b) f() = 3 + f( ) = 3( ) + = 5 c) f() = 3 + f() = 3() + = f() = 9 8 f() = 8 f() = 3 + f( ) = 5 7 MHR Chapter 3

8 Ke Concepts A function is a set of ordered pairs in which, for ever, there is onl one. If an vertical line passes through more than one point on the graph of a relation, then the relation is not a function. The set of the first elements in a relation is called the domain. The set of the second elements in a relation is called the range. An equation that is a function can be named using function notation. In function notation, the smbol f() is another name for and represents the value of the function f at. Communicate Your Understanding. a) Is ever function a relation? Eplain using eamples. b) Is ever relation a function? Eplain using eamples.. Describe how ou would determine if each of the following relations is a function. a) {(, ), (, 3), (, ), (, 5), (, )} b) {(, ), (, ), (, ), (, ), (, )} 3. Describe how ou would determine if each graph models a function. a) b). Describe how ou would determine the domain and range of each of the following relations. a) {(, 3), (3, ), (, ), (5, 3)} b) = Describe how ou would evaluate f() for the function f() = Functions MHR 77

9 Practise A. Determine if each relation is a function. a) b) c) d) e) f) State the domain and range of each relation. a) b) c) d). State the domain and range of each relation. a) {(, 5), (, ), (, 7), (3, 8)} b) {(, ), (, 3), (, 5)} c) {(, ), (, ), (, ), (, ), (, )} d) {(, ), (, ), (3, ), (, ), (7, )} e) f) 8 78 MHR Chapter 3

10 g) h). a) Graph the equation = 3. b) Is the relation a function? 5. Determine if each relation is a function. a) = b) = c) + = 5. If f() = 5, find a) f(8) b) f(5) c) f() d) f() e) f( ) 7. If g() = 3 +, find a) g() b) g() c) g( ) d) g( 3) e) g(.5) 8. If f() = +, find a) f() b) f(5) c) f( ) d) f(.5) e) f(.5) 9. If h() = 3 +, find a) h() b) h() c) h( 3) d) h(.5) e) h(5). If f(n) = n + 5, find a) f() b) f() c) f() d) f( 3) e) f(.5) Appl, Solve, Communicate. List the ordered pairs of the function f() = 7 when the domain is {,,, 5}.. If f() = +, find the value of when the value of f() is a) b) 7 c) 53 d) 9 e) 3. If the domain and range of a relation each contain eactl one real number, describe the graph of the relation.. Cost The cost, C dollars, of purchasing one tpe of ballpoint pen is related to the number of pens purchased, p, b the equation C =.5p. a) Identif the dependent variable and the independent variable. b) Is the cost a function of the number of pens purchased? Eplain. B 5. Determine if each of the following relations is a function. If so, state the dependent variable and the independent variable. a) the time it takes to drive km and the speed of the car b) the ages of students and the numbers of CDs the own c) the number of tickets sold for a school pla at $8 per ticket and the revenue from ticket sales 3. Functions MHR 79

11 . If a point is on the f()-ais of a coordinate grid, what is the -coordinate of the point? Eplain. 7. If the graph of a relation is a vertical line, is the relation a function? Eplain. 8. Find the range of each function when the domain is {,,.5, 3}. a) f() = b) f() = Application Mario sells home theatre sstems. He is paid a weekl salar, plus commission on his sales. His weekl earnings, E(s) dollars, can be determined from his weekl sales, s dollars, using the following functions. When his weekl sales are $3 or less: E(s) =.5s + When his weekl sales are over $3: E(s) =.s + a) Interpret each function in words. b) Identif the dependent variable and the independent variable in each equation. c) State the domain and range of each function. d) What are Mario s weekl earnings when his weekl sales are $? $5?. Measurement The area of a circle, A(r), is a function of the radius, r, where A(r) = πr. State the domain and range of the function.. a) Determine the domain and range of the relation + =. b) Is the relation a function?. Discount prices Neeru is holding a ear-end clearance sale in her clothing store. All prices are discounted b 5%. a) Write an equation that epresses the sale price of an item as a function of its original price. b) If the sale price of a shirt is $, what was its original price? 3. Measurement a) Write an equation that epresses the surface area of a cube as a function of its edge length. b) Determine the surface area of a cube with an edge length of.5 cm.. Inquir/Problem Solving Natalia wants to build a rectangular corral with the largest possible area for her horses. One side of the corral will be the barn wall. She has m of Barn fencing to build the other three sides. a) Write an equation that epresses the area of the corral as a function of the width of the corral. b) Find the maimum area of the corral and the dimensions that give this area. Corral 8 MHR Chapter 3

12 5. Communication Does the graph of 3 = model a function? Eplain.. Algebra If f() = +, find the value(s) of if the value of f() is a) 5 b) c) d) 3 C 7. a) Sketch a graph of a function that has all real numbers in its domain and all real numbers less than or equal to 3 in its range. b) Sketch a graph of a relation that is not a function and that has all real numbers in its domain and all real numbers less than or equal to 3 in its range. 8. Jogging Chelsea jogs several laps around a running track at a stead speed. a) Is the distance she covers a function of the time for which she jogs? Eplain. b) Is her distance from her starting point a function of the time for which she jogs? Eplain. f() f() 9. Describe how the value of is related to the graph of f() = Algebra Solve the following sstem algebraicall. f() = 3 f() = Measurement Epress the area of a circle as a function of its circumference. 3. Algebra a) If f() = + 3, write and simplif f(a). b) If f() = 3, write and simplif f(n + ). c) If g() = +, write and simplif g(m ). d) If f() = 3, write and simplif f(k + ). e) If g() = +, write and simplif g(3t ). f) If f() = 3 +, write and simplif f(3 w). 3. Functions MHR 8