To determine the nature of the function [tex]\( f(x) = (x^m + 9)^2 \)[/tex], we need to check whether it is even, odd, or neither. Let's go through this step by step.
1. Definition of even and odd functions:
- A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
- A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
2. Given function:
[tex]\[ f(x) = (x^m + 9)^2 \][/tex]
3. Check for even function:
- Compute [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = ((-x)^m + 9)^2 \][/tex]
- If [tex]\( m \)[/tex] is even, [tex]\((-x)^m = x^m\)[/tex]. Therefore:
[tex]\[ f(-x) = (x^m + 9)^2 = f(x) \][/tex]
This means that [tex]\( f(x) \)[/tex] is even when [tex]\( m \)[/tex] is even.
4. Check for odd function:
- For [tex]\( f(x) \)[/tex] to be odd, [tex]\( f(-x) \)[/tex] should equal [tex]\(-f(x)\)[/tex].
- However, when [tex]\( m \)[/tex] is odd, [tex]\((-x)^m = -x^m\)[/tex]. Thus:
[tex]\[ f(-x) = (-x^m + 9)^2 \][/tex]
This expression does not simplify to [tex]\(- (x^m + 9)^2 \)[/tex], nor does it simplify directly to [tex]\( (x^m + 9)^2 \)[/tex].
From these steps, we can conclude:
- [tex]\( f(x) \)[/tex] is an even function when [tex]\( m \)[/tex] is even.
- [tex]\( f(x) \)[/tex] is neither even nor odd when [tex]\( m \)[/tex] is odd.
Therefore, the correct statement about [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{f(x)\text{ is an even function for all even values of } m.} \][/tex]
