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Chapter 2: Problem 18

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

Short Answer

Expert verified
The function \(f(x)=x^{3}-x\) is odd.

Step by step solution

01

Testing for an Even Function

Replace \(x\) in the function \(f(x)=x^{3}-x\) with \(-x\) to see if we get \(f(x)\). The new function becomes \(f(-x)=(-x)^{3}-(-x)= -x^{3} + x\). This is not equal to the original function, \(f(x)\), so the function is not even.
02

Testing for an Odd Function

Now, to check if the function is odd, we need to see if \(f(-x)\) is equal to \(-f(x)\). Taking the negative of the original function, we get \(-f(x)=-x^{3}+x\). Comparing this to \(f(-x)\) which is \(-x^{3}+x\), they are the same, implying that the function is odd.

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