Problem 42 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(h(-x) = h(x)\).

Step 2: 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.

Step 3: 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.

Step 4: 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.