To determine whether a function [tex]\( f(x) \)[/tex] is an odd function, we need to check if [tex]\( f(-x) = -f(x) \)[/tex]. Let's go through the steps for [tex]\( f(x) = 9 - 4x^2 \)[/tex]:
1. First, find the expression for [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 9 - 4(-x)^2
\][/tex]
Simplifying this expression:
[tex]\[
f(-x) = 9 - 4(x^2) = 9 - 4x^2
\][/tex]
2. Next, find the negative of the original function, [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -(9 - 4x^2) = -9 + 4x^2
\][/tex]
3. Compare [tex]\( f(-x) \)[/tex] and [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = 9 - 4x^2
\][/tex]
[tex]\[
-f(x) = -9 + 4x^2
\][/tex]
Clearly, [tex]\( f(-x) \neq -f(x) \)[/tex]. Thus, the function [tex]\( f(x) = 9 - 4x^2 \)[/tex] is not an odd function.
Among the given statements, the third one is the one that best describes how to determine whether [tex]\( f(x) = 9 - 4x^2 \)[/tex] is an odd function:
Determine whether [tex]\( 9-4(-x)^2 \)[/tex] is equivalent to [tex]\( -\left(9-4 x^2\right) \)[/tex].
This statement aligns with the requirement to check if [tex]\( f(-x) = -f(x) \)[/tex].
