Answer:
- not even; f(-x) ≠ f(x)
- not odd; f(-x) ≠ -f(x)
Explanation:
The definitions of even and odd functions are cast in terms of the relationships between f(x), -f(x), and f(-x). That is why you're asked to examine those functions here.
For a polynomial, the "even" or "odd" status of a function depends on the exponents of the variable. If they are all even, the polynomial function is even. If they are all odd, the polynomial function is odd. If they are mixed, then the function is neither even nor odd.
Short answer:
Here, the degrees of the two terms are 0 (even) and 3 (odd). The function f(x) is neither even nor odd.
Long answer:
Simplifying the various transformations, we find ...
f(-x) = 25 -(-x)³ = 25 +x³ ≠ f(x)
-f(x) = -25 +x³ ≠ f(-x)
The function is not even, since f(-x) ≠ f(x).
The function is not odd, since f(-x) ≠ -f(x).