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Chapter 1: Problem 72

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$g(x)=x^{3}-5 x$$

Short Answer

Expert verified
The function \(g(x)=x^{3}-5x\) is odd and is symmetric about the origin.

Step by step solution

01

Calculate \(g(-x)\)

Replace \(x\) with \(-x\) in the function and simplify. So, \(g(-x) = (-x)^{3}-5(-x) = -x^{3}+5x\).
02

Compare \(g(-x)\) and \(g(x)\)

Check if \(g(-x)\) is equal to \(g(x)\) or \(-g(x)\). However, \(g(-x) = -x^{3}+5x\) and \(g(x) = x^{3}-5x\) are not equal. By negating \(g(x)\), we get \(-g(x) = -x^{3}+5x\), which is equal to \(g(-x)\), hence the function is odd.
03

Describe the Symmetry

Since the function is odd, it is symmetric about the origin. This means if you rotate the graph 180 degrees around the origin, it remains unchanged.

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