Problem 45 - 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. $$f(x)=\frac{1}{5} x^{6}-3 x^{2}$$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{1}{5} x^{6}-3 x^{2}\) is an even function and its graph is symmetric with respect to the y-axis.

Step 1: Define an Even Function

An even function is a function for which the following equation is true: \(f(x) = f(-x)\) for any value of \(x\) in the domain of \(f\). Even functions are symmetric about the y-axis.

Step 2: Test for Even Function

Plug in \(-x\) into the function instead of \(x\): \(f(-x)=\frac{1}{5} (-x)^{6}-3 (-x)^{2} = \frac{1}{5} x^{6}-3 x^{2}\). We can see that \(f(-x)\) is equal to \(f(x)\). Thus, \(f(x)\) is an even function.

Step 3: Define an Odd Function

An odd function is a function for which the following equation is true: \(f(-x) = -f(x)\) for any \(x\) in the domain of \(f\). Odd functions are symmetric about the origin.

Step 4: Test for Odd function

We have already established that the function \(f(x)\) is an even function, so it cannot be an odd function. There is no need to test for this.

Step 5: Symmetry of the Graph

Since this function is an even function, its graph will be symmetric about the y-axis.