We are given the function $f(x) = x + x^2 + x^5$ and asked to determine whether it is even, odd, or neither.

1. Problem Description

We are given the function $f(x) = x + x^2 + x^5$ and asked to determine whether it is even, odd, or neither.

2. Solution Steps

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.

We are given $f(x) = x + x^2 + x^5$.

Let's find $f(-x)$:

$f(-x) = (-x) + (-x)^2 + (-x)^5 = -x + x^2 - x^5$.

Now we check if $f(-x) = f(x)$.

$f(x) = x + x^2 + x^5$

$f(-x) = -x + x^2 - x^5$

Since $x + x^2 + x^5 \neq -x + x^2 - x^5$, $f(x)$ is not an even function.

Next, we check if $f(-x) = -f(x)$.

$-f(x) = -(x + x^2 + x^5) = -x - x^2 - x^5$

$f(-x) = -x + x^2 - x^5$

Since $-x - x^2 - x^5 \neq -x + x^2 - x^5$, $f(x)$ is not an odd function.

Therefore, $f(x)$ is neither even nor odd.

3. Final Answer

The function $f(x)$ is neither odd nor even.