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Chapter 1: Problem 38

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

Short Answer

Expert verified
The function \(f(x) = x^{3} - x\) is odd, and its graph is symmetric about the origin.

Step by step solution

01

Determine if the function is even

To determine whether the function \(f(x) = x^{3} - x\) is even, substitute \(-x\) for \(x\) and simplify: \(f(-x) = (-x)^{3} -(-x) = -x^{3} + x\). This does not equal to \(f(x)\), so the function is not even.
02

Determine if the function is odd

To determine whether the function \(f(x) = x^{3} - x\) is odd, compare the value of \(f(-x) = -x^{3} + x\) found in the first step to \(-f(x)\). Since \(-f(x) = -(x^{3} - x) = -x^{3} + x\), and \(f(-x) = -f(x)\), the function is odd.
03

Determine the graph symmetry

As we have established that the function is odd, by definition its graph will show symmetry about the origin. This means if the portion of the graph to the left of the y-axis was flipped over both the x and y axis, it would coincide with the portion of the graph to the right of the y-axis.

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