Problem 64 - College Algebra

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: Check for 'Even' Property

Firstly, we substitute \(x\) with \(-x\) in the function which yields \(g(-x) = (-x)^{2} - (-x) = x^{2}+x\). This is not equal to \(g(x)\), so the function is not even.

Step 2: Check for 'Odd' Property

Now, consider if our function is an odd function. We require \(g(-x) = -g(x)\). From 'Step 1', we know that \(g(-x) = x^{2}+x\), and we calculate \(-g(x) = -(x^{2} - x) = -x^{2} + x\). Clearly, \(g(-x) \neq -g(x)\), showing that the function is not odd.

Step 3: Conclusion

As the function fulfills neither the even nor the odd property, it can be concluded that the function \(g(x)=x^{2}-x\) is neither even nor odd.