To determine whether the function [tex]\( f(x) = (x^m + 9)^2 \)[/tex] is even or odd, we need to examine how [tex]\( f(x) \)[/tex] behaves when we replace [tex]\( x \)[/tex] with [tex]\(-x\)[/tex]:
[tex]\[
f(-x) = \left((-x)^m + 9\right)^2
\][/tex]
- For [tex]\( f(x) \)[/tex] to be an even function, [tex]\( f(-x) \)[/tex] must be equal to [tex]\( f(x) \)[/tex] for all [tex]\( x \)[/tex].
- For [tex]\( f(x) \)[/tex] to be an odd function, [tex]\( f(-x) \)[/tex] must be equal to [tex]\( -f(x) \)[/tex] for all [tex]\( x \)[/tex].
We will explore both cases.
### Case 1: Even function
To check if [tex]\( f(x) \)[/tex] is even, let's simplify [tex]\( f(-x) \)[/tex] and compare it to [tex]\( f(x) \)[/tex]:
[tex]\[
f(-x) = \left((-x)^m + 9\right)^2
\][/tex]
For [tex]\( f(x) \)[/tex] to be even, we need:
[tex]\[
\left((-x)^m + 9\right)^2 = \left(x^m + 9\right)^2
\][/tex]
This is only true if:
[tex]\[
(-x)^m + 9 = x^m + 9
\][/tex]
This further simplifies to:
[tex]\[
(-x)^m = x^m
\][/tex]
This equation holds if and only if [tex]\( m \)[/tex] is even. Therefore, [tex]\( f(x) \)[/tex] is even when [tex]\( m \)[/tex] is even.
### Case 2: Odd function
To check if [tex]\( f(x) \)[/tex] is odd, we need to see if [tex]\( f(-x) = -f(x) \)[/tex]:
[tex]\[
f(-x) = \left((-x)^m + 9\right)^2
\][/tex]
For [tex]\( f(x) \)[/tex] to be odd, we need:
[tex]\[
\left((-x)^m + 9\right)^2 = -\left(x^m + 9\right)^2
\][/tex]
Since the square of any real number is always non-negative, [tex]\((-x)^m + 9\right)^2 and \left(x^m + 9\right)^2 are always non-negative. Hence, equating \( \left((-x)^m + 9\right)^2 \)[/tex] to the negative of [tex]\( \left(x^m + 9\right)^2 \ is only possible if both sides are zero, which is an impossibility unless \( x^m + 9 = 0 \)[/tex] for all [tex]\( x \)[/tex], which does not generally hold.
Therefore, the function [tex]\( f(x) = (x^m + 9)^2 \)[/tex] cannot be an odd function for any value of [tex]\( m \)[/tex].
Given these cases, we conclude that:
- [tex]\( f(x) \)[/tex] is an even function for all even values of [tex]\( m \)[/tex].
Hence, the correct statement about [tex]\( f(x) = (x^m + 9)^2 \)[/tex] is:
- [tex]\( f(x) \)[/tex] is an even function for all even values of [tex]\( m \)[/tex].
