Problem 21 - College Algebra Essentials

Question

Determine whether each function is even, odd, or neither. $$h(x)=x^{2}-x^{4}$$

Step-by-Step Solution

Answer

The function \(h(x) = x^{2} - x^{4}\) is an even function.

Step 1: Identifying the Function

The function given is \(h(x) = x^{2} - x^{4}\). The goal is to determine whether this function is even, odd, or neither.

Step 2: Substitute -x for x

Substitute -x for x in the function \(h(x)\) to establish whether the function is even or odd. In the case of an even function, the result will be equal to \(h(x)\) (i.e. \(h(-x) = h(x)\)), and for an odd function, it'll be equal to \(-h(x)\) (i.e. \(h(-x) = -h(x)\)). Thus, when -x is substituted into the function \(h(x)\), it yields \(h(-x) = (-x)^{2} - (-x)^{4} = x^{2} - x^{4}\).

Step 3: Comparing the original function to h(-x)

Now, it can be observed that the result of substituting -x for x (\(h(-x) = x^{2} - x^{4}\)) is identically equal to the original function \(h(x) = x^{2} - x^{4}\), which satisfies the condition of an even function (i.e., \(h(-x) = h(x)\)). The function does not meet the criteria of an odd function since \(h(-x) \neq -h(x)\).