Question

In Exercises \(21-30\) , determine whether the function is even, odd, or neither. Try to answer without writing anything (except the answer). $$y=x^{2}-3$$

Short answer

The function \(y=x^{2}-3\) is even.

Step-by-step solution

Test for Even Function

Verify if the function is even. That is, check whether the expression \(f(-x)\) will result in the original function \(f(x)\). For the given function \(y=x^{2}-3\), if we replace \(x\) with \(-x\), we get \((-x)^{2}-3 = x^{2}-3\), which is the original function again. Therefore, this function is even.

Test for Odd Function

For completeness, check whether the function is odd. In other words, investigate if \(-f(x)\) equals \(f(-x)\). For our given function, replacing \(x\) with \(-x\) in \(-f(x)\) yields \(-((-x)^{2}-3) = -(x^{2}-3)\). This does not yield the same result as \(f(-x)\), which we found earlier to be \(x^{2}-3\). Hence, the function is not odd.