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Chapter 1: Problem 46

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.

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