Problem 20 - College Algebra Essentials

Question

Determine whether each function is even, odd, or neither. $$g(x)=x^{2}-x$$

Step-by-Step Solution

Answer

The function \(g(x)=x^{2}-x\) is neither even nor odd.

Step 1: Analyze the function

First, consider the function \(g(x)=x^{2}-x\). This is a polynomial function of degree 2.

Step 2: Test for evenness

To check if it is an even function, we replace \(x\) by \(-x\) in the function and simplify. This gives \(g(-x)=(-x)^{2}-(-x)=x^{2}+x\). Comparing this with the original function \(g(x)=x^{2}-x\), it's clear that \(g(x) ≠ g(-x)\), therefore, the function is not even.

Step 3: Test for oddness

To check if it is an odd function, we need to compare \(g(x)\) and \(-g(-x)\). We already have \(g(-x)\) from Step 2, so \(-g(-x)=-(x^{2}+x)=-x^{2}-x\). Comparing this to the original function \(g(x)=x^{2}-x\), it can be seen that \(g(x) ≠ -g(-x)\), therefore the function is not odd.