66 CHAPTER 2 Functions and Their Graphs CAUTION The notation ( ) = y f x denotes a function f. It does NOT mean “f times x.” j Figure 14 illustrates some other functions. Notice that in every function, for each x in the domain, there is one corresponding value in the range. Figure 14 Domain Range 1 21 0 2 1 5 f(1) 5 f(21) 0 5 f(0) 2 5 f( 2 ) x f(x) 5 x 2 (a) f(x) 5 x 2 Domain Range 22 21 4 5 F(22) 21 5 F(21) 5 F(4) x F(x) 5 (b) F(x) 5 1 –x 1 –x 1 – 2 1 – 4 2 Domain Range 0 1 2 4 0 5 g(0) 1 5 g(1) 2 5 g(2) 2 5 g(4) x g(x) 5 x (c) g(x) 5 x Domain Range 0 22 3 3 5 G(0) 5 G(22) 5 G(3) x G(x) 5 3 (d) G(x) 5 3 For a function ( ) = y f x , the variable x is called the independent variable because it can be assigned any number from the domain. The variable y is called the dependent variable because its value depends on x. Any symbol can be used to represent the independent and dependent variables. For example, if f is the cube function, then f can be given by ( ) = f x x3 or ( ) = f t t 3 or ( ) = f a a .3 All three functions are the same. Each says to cube the independent variable to get the output. In practice, the symbols used for the independent and dependent variables are based on common usage, such as using t for time and a for acceleration. The independent variable is also called the argument of the function. Thinking of the independent variable as an argument can sometimes make it easier to find the value of a function. For example, if f is the function defined by ( ) = f x x ,3 then f tells us to cube the argument. Then ( ) f 2 means to cube 2, ( ) f a means to cube the number a, and ( ) + f x h means to cube the quantity +x h. Finding Values of a Function For the function f defined by ( ) = − f x x x 2 3 , 2 evaluate (a) ( ) f 3 (b) ( ) ( ) + f x f 3 (c) ( ) f x 3 (d) ( ) − f x (e) ( ) −f x (f) ( ) f x3 (g) ( ) + f x 3 (h) ( ) + f x h Solution EXAMPLE 6 (a) Substitute 3 for x in the equation for ( ) = − f f x x x , 2 3 , 2 to get ( ) = ⋅ − ⋅ = − = f 3 2 3 3 3 18 9 9 2 The image of 3 is 9. (b) ( ) ( ) ( ) + = − + = − + f x f x x x x 3 2 3 9 2 3 9 2 2 (c) Multiply the equation for f by 3. ( ) ( ) = − = − f x x x x x 3 3 2 3 6 9 2 2
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