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Chapter 2: Problem 68

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

Short Answer

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

Step by step solution

01

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 \(f(-x) = 2x^2 + x^4 + 1\).
02

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.
03

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.

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