Question

What type of function is g(x)=x−−√3 ?(1 point) Responses even even odd odd neither even nor odd neither even nor odd both even and odd

Answer 1

To determine whether the function \( g(x) = \sqrt{x}^3 \) is even, odd, or neither, we need to check the definitions of even and odd functions:

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 \( x \).

Now, let's analyze the function \( g(x) = \sqrt{x}^3 \).

First, we simplify the expression:

\[
g(x) = (\sqrt{x})^3 = x^{3/2}.
\]

Next, we find \( g(-x) \):

\[
g(-x) = \sqrt{-x}^3.
\]

However, \( \sqrt{-x} \) is not defined for real numbers when \( x \) is positive (making the domain restrictive). Thus, we can only evaluate \( g(-x) \) for \( x < 0 \).

For negative \( x \):
- Since \( g(-x) \) involves taking the square root of a negative number, \( g(-x) \) is not defined in the set of real numbers.
- Since \( g(x) \) is not defined for negative \( x \), we cannot demonstrate either the evenness or oddness of \( g(x) \).

Thus, we conclude that \( g(x) \) is neither even nor odd.

Final answer: **neither even nor odd**.