The problem asks to determine if the function $f(x) = x^5 + x$ is an even function. The solution provided only considers $f(x) = x^5$.

1. Problem Description

The problem asks to determine if the function $f(x) = x^5 + x$ is an even function. The solution provided only considers $f(x) = x^5$.

2. Solution Steps

To determine if a function is even, we need to check if $f(-x) = f(x)$ for all $x$.

Given the function $f(x) = x^5 + x$, we need to find $f(-x)$.

Substitute $-x$ for $x$ in the function:

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

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

Now, compare $f(-x)$ to $f(x)$.

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

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

Since $f(-x) = -f(x)$, the function is an odd function. For a function to be even, $f(-x) = f(x)$, which is not the case here. The original problem only looks at $x^5$ which is odd. However, that is incomplete and does not answer the question of whether $x^5 + x$ is even.

Even Function: $f(-x) = f(x)$

Odd Function: $f(-x) = -f(x)$

3. Final Answer

The function $f(x) = x^5 + x$ is an odd function, not an even function.