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Chapter 1: Problem 42

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

Short Answer

Expert verified
The function \(h(x)=2 x^{2}+x^{4}\) is an even function and its graph is symmetric with respect to the y-axis but not with respect to the origin.

Step by step solution

01

Test for even function

Plugging \(-x\) into the function gives \(h(-x) = 2(-x)^2 + (-x)^4 = 2x^2 + x^4 =h(x)\), which confirms that \(h(x)\) is an even function because \(h(-x) = h(x)\).
02

Test for odd function

Plugging \(-x\) into the function gives \(h(-x) = -h(x) = -2x^2 - x^4\), but this is not equal to \(h(x)\), hence the function is not odd.
03

Test for y-axis symmetry

The function is symmetric with respect to the y-axis because every point \((x, y)\) on the graph has a corresponding point \((-x, y)\). This follows from the fact that \(h(x)\) is an even function.
04

Test for origin symmetry

The function is not symmetric with respect to the origin because not every \((x, y)\) on the graph has a corresponding point \((-x, -y)\). This follows from the fact that \(h(x)\) is not an odd function.

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Most popular questions from this chapter

Give an example of a circle's equation in standard form. Describe how to find the center and radius for this circle.

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