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Chapter 1: Problem 37

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 with respect to the origin.

Step by step solution

01

Determine if function is even

Replace \( x \) with \(-x\) in the function \(f(x)=x^{3}+x\) to get \(f(-x)=(-x)^{3}+(-x)=-x^3-x\), which does not equal the original function. Hence, the function is not even.
02

Determine if function is odd

Since the function \(f(x)=x^{3}+x\) is not even, check if it is odd by checking if \(f(-x)\) equals \(-f(x)\). For the function, \(-f(x)=-x^3-x\), which equals \(f(-x)=-x^3-x\), meaning the function \(f(x)=x^{3}+x\) is odd.
03

Determine symmetry

The graph of an odd function is symmetric with respect to the origin. Therefore, the graph of the function \(f(x)=x^{3}+x\) is symmetric with respect to the origin.

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