To determine if the function $f(h(g)) = x^5 + x^4$ is even, odd, or neither, we need to analyze the function's behavior when substituting $-x$ for $x$.

Definitions

1. Even Function: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$.
2. Odd Function: A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$.
3. Neither: If a function does not satisfy either of these conditions, it is neither even nor odd.

Given Function

The function is: $f(x) = x^5 + x^4$

Step-by-Step Analysis

1. Substitute $-x$ into the function: $f(-x) = (-x)^5 + (-x)^4$

2. Simplify the expression: $(-x)^5 = -x^5 \quad \text{(since raising to an odd power preserves the sign)}$ $(-x)^4 = x^4 \quad \text{(since raising to an even power results in a positive number)}$ Therefore, $f(-x) = -x^5 + x^4$

3. Compare $f(-x)$ with $f(x)$: $f(x) = x^5 + x^4, \quad f(-x) = -x^5 + x^4$

   • $f(-x) \neq f(x)$, so the function is not even.
   • $f(-x) \neq -f(x)$ because: $-f(x) = -(x^5 + x^4) = -x^5 - x^4$ And $f(-x) = -x^5 + x^4$, which is different from $-f(x)$.

Conclusion

The function $f(x) = x^5 + x^4$ is neither even nor odd.

Would you like more details on the topic, or do you have any other questions?

Here are some related questions to consider:

1. What are examples of even and odd functions?
2. How do you determine if a polynomial function is even or odd?
3. What are the properties of even and odd functions with respect to their graphs?
4. How do trigonometric functions relate to the concepts of even and odd functions?
5. Can a function be both even and odd?

Tip: For any function, checking whether it is even or odd helps to understand its symmetry properties, which is useful for graphing and analyzing functions.