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Chapter 1: Problem 41

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. $$h(x)=x^{2}-x^{4}$$

Short Answer

Expert verified
The function \(h(x) = x^{2}-x^{4}\) is an even function with a graph that is symmetric with respect to the \(y\)-axis.

Step by step solution

01

Check if the function is even

Substitute \(x\) with \(-x\) in the equation, so \(h(-x) = (-x)^{2}-(-x)^{4}\). This simplifies to \(h(-x) = x^{2}-x^{4}\), which is the same as the original equation, so \(h(x) = h(-x)\). Therefore, the function is even.
02

Check if the function is odd

Substitute \(x\) with \(-x\) in the equation, so \(h(-x) = (-x)^{2}-(-x)^{4}\). This simplifies to \(h(-x) = x^{2}-x^{4}\), which doesn't satisfy the condition \(f(-x) = -f(x)\). Therefore, the function is not odd.
03

Determine the symmetry of the graph

Since the function is even, its graph is symmetric with respect to the \(y\)-axis and not symmetric with respect to the origin.

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