To determine whether the function [tex]\( f(x) = \left(x^m + 9\right)^2 \)[/tex] is even, odd, or neither for different values of [tex]\( m \)[/tex], we need to check the symmetry properties of the function.
### Definitions
1. Even function: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex].
2. Odd function: A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex].
3. Neither: A function that does not satisfy either of the above properties.
### Steps to Determine the Nature of [tex]\( f(x) \)[/tex]
1. Original function:
[tex]\[
f(x) = \left(x^m + 9\right)^2
\][/tex]
2. Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = \left((-x)^m + 9\right)^2
\][/tex]
3. Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
- For even [tex]\( m \)[/tex] (where [tex]\( m \)[/tex] is an even number):
[tex]\[
(-x)^m = x^m
\][/tex]
Therefore,
[tex]\[
f(-x) = \left(x^m + 9\right)^2 = f(x)
\][/tex]
This indicates that [tex]\( f(x) \)[/tex] is an even function when [tex]\( m \)[/tex] is even.
- For odd [tex]\( m \)[/tex] (where [tex]\( m \)[/tex] is an odd number):
[tex]\[
(-x)^m = -x^m
\][/tex]
Therefore,
[tex]\[
f(-x) = \left(-x^m + 9\right)^2
\][/tex]
Since this expression does not simplify to [tex]\( f(x) \)[/tex] or [tex]\(-f(x)\)[/tex] in general, [tex]\( f(x) \)[/tex] is neither even nor odd for odd [tex]\( m \)[/tex].
### Conclusion
From the analysis above:
- [tex]\( f(x) \)[/tex] is an even function for all even values of [tex]\(m\)[/tex].
- [tex]\( f(x) \)[/tex] is not an odd function for any values of [tex]\( m \)[/tex].
Thus, the correct statement is:
[tex]\[ f(x) \text{ is an even function for all even values of } m. \][/tex]
