Problem 37
Question
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. $$f(x)=x^{3}+x$$
Step-by-Step Solution
Verified Answer
The function \(f(x)=x^{3}+x\) is odd, and its graph is symmetric with respect to the origin.
1Step 1: 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.
2Step 2: 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.
3Step 3: 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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Problem 37
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