Problem 43 - Precalculus Essentials

Question

Determine whether each function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the \(y\) -axis, the origin, or neither. $$f(x)=x^{2}-x^{4}+1$$

Step-by-Step Solution

Answer

The function \(f(x)=x^{2}-x^{4}+1\) is even and its graph is symmetric with respect to the y-axis.

Step 1: Determine if the function is even

Replace \(x\) with \(-x\) in the function \(f(x)\) and simplify. If the result is equal to \(f(x)\), then the function is even. Doing so, we get \(f(-x)=(-x)^{2}-(-x)^{4}+1=x^{2}-x^{4}+1=f(x)\). Hence, \(f(x)\) is an even function.

Step 2: Determine if the function is odd

Even though we've already found the function to be even, let's verify if it's odd or not for instructional purposes. Replace \(x\) with \(-x\) in the function \(f(x)\) and simplify. If the result is equal to \(-f(x)\), then the function is odd. Doing so, we get \(f(-x)=(-x)^{2}-(-x)^{4}+1=x^{2}-x^{4}+1=f(x)\), which is not equal to \(-f(x)\). Therefore, \(f(x)\) is not odd.

Step 3: Determine symmetry of the function's graph

An even function's graph is symmetric with respect to the y-axis, whereas an odd function's graph is symmetric with respect to the origin. Since we determined earlier that \(f(x)\) is an even function, thus its graph is symmetric with respect to the y-axis.