Chapter 1 | Functions and Graphs
27
b. f ( x ) =2 x 5 −4 x +5 c. f ( x ) = 3 x x 2 +1
Solution To determine whether a function is even or odd, we evaluate f (− x ) and compare it to f ( x ) and − f ( x ). a. f (− x ) =−5(− x ) 4 +7(− x ) 2 −2=−5 x 4 +7 x 2 −2= f ( x ). Therefore, f is even. b. f (− x ) =2(− x ) 5 −4(− x )+5=−2 x 5 +4 x +5. Now, f (− x ) ≠ f ( x ). Furthermore, noting that − f ( x ) =−2 x 5 +4 x −5, we see that f (− x ) ≠− f ( x ). Therefore, f is neither even nor odd. c. f (− x ) =3(− x )/((− x ) 2 +1}=−3 x /( x 2 +1) =−[3 x /( x 2 +1)] =− f ( x ). Therefore, f is odd.
Determine whether f ( x ) =4 x 3 −5 x is even, odd, or neither.
1.7
One symmetric function that arises frequently is the absolute value function , written as | x |. The absolute value function is defined as (1.2) f ( x ) = Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if x <0, then | x | =− x >0, and if x >0, then | x | = x >0. However, for x =0, | x | =0. Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if x =0, theoutput | x | =0. We conclude that the range of the absolute value function is ⎧ ⎩ ⎨ y | y ≥0 ⎧ ⎩ ⎨ − x , x <0 x , x ≥0 . ⎫ ⎭ ⎬ . In Figure 1.14 , we see that the absolute value function
is symmetric about the y -axis and is therefore an even function.
Figure 1.14 The graph of f ( x ) = | x | is symmetric about the y -axis.
Made with FlippingBook - professional solution for displaying marketing and sales documents online