Problem 66 - College Algebra

Question

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

Step-by-Step Solution

Answer

The function \(h(x) = 2x^2 + x^4\) is an even function.

Step 1: Definition of an Even Function

A function \(f(x)\) is called even if for every \(x\) in the function's domain, \(f(x) = f(-x)\). In other words, replacing every instance of \(x\) with \(-x\) in the function and simplifying should yield the original function.

Step 2: Check if \(h(x)\) is Even

Substitute \(-x\) for \(x\) in \(h(x) = 2x^{2} + x^{4}\) to get \(h(-x) = 2(-x)^{2} + (-x)^{4}\). Simplifying, you get \(h(-x) = 2x^2 + x^4\), which is the equivalent to the original function \(h(x)\). Therefore, \(h(x)\) is an even function.

Step 3: Definition of an Odd Function

A function \(f(x)\) is called odd if for every \(x\) in the function's domain, \(f(x) = -f(-x)\). In other words, replacing every instance of \(x\) with \(-x\) in the function and simplifying should yield the negative of the original function.

Step 4: Check if \(h(x)\) is Odd

For completeness, check if \(h(x)\) may also be odd. Substitute \(-x\) for \(x\) in \(h(x) = 2x^{2} + x^{4}\) to get \(h(-x) = -(2x^2 + x^4)\). But this is not equivalent to the original function \(h(x)\), so \(h(x)\) is not an odd function.