Final answer:
To determine whether a function is even or odd, substitute -x for x and simplify.
If the function remains unchanged, it is even.
If the function becomes its opposite, it is odd.
h(x) = x^5 + 4x^3 - x is even, while g(x) = -x^4 + 8x^2 - 5 is odd.
Step-by-step explanation:
In order to determine whether a function is even or odd, we need to consider the properties of even and odd functions.
An even function is symmetric about the y-axis, which means that if we reflect the function across the y-axis, it remains unchanged.
This can be represented as f(x) = f(-x).
On the other hand, an odd function is symmetric about the origin, which means that if we reflect the function across the origin, it remains unchanged.
This can be represented as f(x) = -f(-x).
Let's apply these definitions to the given functions:
For h(x) = x^5 + 4x^3 - x:
- To check if it's even, we substitute -x for x and simplify:
h(-x) = (-x)^5 + 4(-x)^3 - (-x)
h(-x) = -x^5 + 4x^3 + x - Since h(x) = h(-x), the function h(x) is even.
For g(x) = -x^4 + 8x^2 - 5:
- To check if it's odd, we substitute -x for x and simplify:
g(-x) = -(-x)^4 + 8(-x)^2 - 5
g(-x) = -x^4 + 8x^2 - 5 - Since g(x) = -g(-x), the function g(x) is odd.