Problem 68 - College Algebra

Question

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

Step-by-Step Solution

Answer

The function \(f(x)=2 x^{2}+x^{4}+1\) is even.

Step 1: Substitute -x into the function

Begin by substituting \(x\) with \(-x\) in the function. The equation then becomes \(f(-x) = 2(-x)^2 + (-x)^4 + 1\). Simplify this to \(f(-x) = 2x^2 + x^4 + 1\).

Step 2: Comparison with the original function

As we compare \(f(x)\) with \(f(-x)\), we see that they are equal. So, \(f(x) = f(-x)\) for all x in the function's domain.

Step 3: Test for Oddness

-f(-x) equals \(-f(x)\) for an odd function. Hence, \(-f(-x) = -(2x^2 + x^4 + 1)\) which simplifies to \(-2x^2 - x^4 - 1\). This is not equal to \(f(x)\), so the function is not odd.