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Chapter 1: Problem 44

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. $$f(x)=2 x^{2}+x^{4}+1$$

Short Answer

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

Step by step solution

01

Determine Type of Function

Substitute \(-x\) into the function \(f(x)=2 x^{2}+x^{4}+1\). If the original function is produced, then it's even, if the negation of the original function is produced then it's odd. Upon replacement, the function becomes \(f(-x) = 2(-x)^{2}+(-x)^{4}+1 = 2x^{2}+x^{4}+1\). Therefore, the given function is an even function as \(f(x) = f(-x)\).
02

Determine Symmetry of Graph

As derived in the previous step, the function is even which indicates that the graph of the function will be symmetric about the y-axis.

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