Problem 48 - 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. $$f(x)=x^{2} \sqrt{1-x^{2}}$$

Step-by-Step Solution

Answer

The function \(f(x) = x^{2} \sqrt{1 - x^{2}}\) is an even function, and its graph is symmetric with respect to the y-axis.

Step 1: Define the Function and Determine its Type

First, note that a function \(f(x)\) is even if \(f(-x) = f(x)\) and is odd if \(f(-x) = -f(x)\). Plugging \(-x\) into our given function, we get \[f(-x) = (-x)^{2} \sqrt{1 - (-x)^{2}} = x^{2} \sqrt{1 - x^{2}}\]This is the same as our original function \(f(x) = x^{2} \sqrt{1 - x^{2}}\), which means that \(f(x)\) is an even function.

Step 2: Determine the Symmetry of the Function

Identify whether the graph of the function is symmetric with respect to the y-axis, the origin, or neither by understanding the properties of symmetry:- A function is symmetric about the y-axis if it is an even function. - A function is symmetric about the origin if it is an odd function.As we have determined that the function is even, it means that its graph is symmetric about the y-axis.