Problem 67 - College Algebra

Question

Determine whether each function is even, odd, or neither. $$f(x)=x^{2}-x^{4}+1$$

Step-by-Step Solution

Answer

The function \(f(x)=x^{2}-x^{4}+1\) is an even function based on our analysis.

Step 1: Substitute -x for x

Substitute -x for x in the function \(f(x)=x^{2}-x^{4}+1\). This gives us \(f(-x)=(-x)^{2}-(-x)^{4}+1\).

Step 2: Simplify the Equation

Simplify \(f(-x)=(-x)^{2}-(-x)^{4}+1\) to \(f(-x) = x^{2}-x^{4}+1\), as squaring or raising to the fourth power (-x) gives back positive values.

Step 3: Compare with Original Function

Compare \(f(-x) = x^{2}-x^{4}+1\) with our original function \(f(x)=x^{2}-x^{4}+1\). We observe that \(f(x) = f(-x)\), which tells us the function is even.