Problem 61 - College Algebra

Question

Determine whether each function is even, odd, or neither. $$f(x)=x^{3}+x$$

Step-by-Step Solution

Answer

The function \(f(x)=x^{3}+x\) is odd.

Step 1: Calculate \(f(-x)\)

Plug \(-x\) into the function \(f(x) = x^{3} + x\), then \(f(-x) = (-x)^{3} + (-x) = -x^{3} - x\).

Step 2: Compare \(f(-x)\) with \(f(x)\) and \(-f(x)\)

Now compare \(f(-x)\) with \(f(x)\) and \(-f(x)\). If \(f(-x) = f(x)\), then the function is even. If \(f(-x) = -f(x)\), then the function is odd. Here we see that, \(f(-x)= -f(x)\), so function is odd.