Problem 65 - College Algebra

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: Find \(h(-x)\)

First, we will substitute \(x\) with \(-x\) in the function \(h(x)\). Doing this, we get \(h(-x)=(-x)^{2}-(-x)^{4}\)

Step 2: Simplify the function \(h(-x)\)

Now, let's simplify the function \(h(-x)\). Remember, the square of any real number, whether positive or negative, is always positive. Thus, \((-x)^{2}=x^{2}\) and \((-x)^{4}=x^{4}\). Substituting these results, we get \(h(-x)=x^{2}-x^{4}\)

Step 3: Compare \(h(x)\) and \(h(-x)\)

Now, let's compare the original function \(h(x)=x^{2}-x^{4}\) and the function we found \(h(-x)=x^{2}-x^{4}\). As we can see, \(h(x) = h(-x)\) for every \(x\). Therefore, the function \(h(x)\) is even