Problem 46 - 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. $$f(x)=2 x^{3}-6 x^{5}$$

Step-by-Step Solution

Answer

The function \(f(x)=2 x^{3}-6 x^{5}\) is neither even nor odd. Its graph is also not symmetric with respect to the y-axis or the origin.

Step 1: Determine if the function is even or odd

A function \(f(x)\) is even if \(f(-x) = f(x)\) and it is odd if \(f(-x) = -f(x)\). Let's check this for \(f(x)=2 x^{3}-6 x^{5}\). We calculate \(f(-x)\), which is \(2 (-x)^{3}-6 (-x)^{5} = -2 x^{3} + 6 x^{5}\). This result is not equal to \(f(x)\) and it's not equal to \(-f(x)\), so the function is neither even nor odd.

Step 2: Check graph symmetry

As the function is neither even nor odd, the function's graph won't be symmetric with respect to the y-axis nor with respect to the origin.