Problem 26 - College Algebra Essentials

Question

Determine whether each function is even, odd, or neither. $$f(x)=2 x^{3}-6 x^{5}$$

Step-by-Step Solution

Answer

The function \(f(x) = 2x^3 - 6x^5\) is neither even nor odd.

Step 1: Write out the Function

First, the function given is \(f(x) = 2x^3 - 6x^5\).

Step 2: Compute \(f(-x)\)

Substitute \(-x\) into the function: \(f(-x) = 2(-x)^3 - 6(-x)^5 = -2x^3 + 6x^5\).

Step 3: Compare \(f(-x)\) and \(f(x)\)

Now compare \(f(-x)\) and \(f(x)\). If they are equal, then the function is even. If \(f(-x) = -f(x)\) then the function is odd. Here, \(f(-x)\) is certainly not equal to \(f(x)\), so the function isn't even. But also, \(f(-x)\) isn't the negative of \(f(x)\), so the function isn't odd either.