Algebraically determine whether the function below is even, odd, or neither.
$$f ( s ) = 8 s ^ { 7 / 6 }$$
A) even
B) odd
C) cannot be determined
D) neither

Question

Quizplus · Accepted Answer

A function is even if $f(-x) = f(x)$ for all $x$ in the domain. A function is odd if $f(-x) = -f(x)$ for all $x$ in the domain.

Let's first check if $f(-s) = f(s)$:

$$f(-s) = 8(-s)^{\frac{7}{6}} = -8s^{\frac{7}{6}}$$

$$f(s) = 8s^{\frac{7}{6}}$$

Since $f(-s) = -f(s)$, the function $f(s)$ is odd. However, if we made the mistake of checking if $f(-s) = -f(s)$ instead, we would get:

$$f(-s) = 8(-s)^{\frac{7}{6}} = -8s^{\frac{7}{6}}$$

$$-f(s) = -8s^{\frac{7}{6}}$$

Since $f(-s) 
eq -f(s)$, the function $f(s)$ would seem to be neither even nor odd. Therefore, the correct answer is D) neither.

Algebraically Determine Whether the Function Below Is Even, Odd, or Neither