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Chapter 2: Problem 69

Determine whether each function is even, odd, or neither. $$f(x)=\frac{1}{5} x^{6}-3 x^{2}$$

Short Answer

Expert verified
The function \(f(x)=\frac{1}{5} x^{6}-3 x^{2}\) is even, not odd.

Step by step solution

01

Check if the function is even

Substitute \(-x\) for \(x\) in the function and simplify. If the result is the same as the original function \(f(x)=f(-x)\), then the function is even. So, we have \(f(-x)=\frac{1}{5} (-x)^{6}-3(-x)^{2}=\frac{1}{5} x^{6}-3 x^{2}=f(x)\). Therefore, the function is even.
02

Check if the function is odd

Though we've already established that the function is even, we'll still check if it is odd for the sake of completion. Substitute \(-x\) for \(x\) in the function and simplify. If the result is equivalent to \(-f(x)\), then the function is odd. So, we have \(f(-x)=\frac{1}{5} (-x)^{6}-3(-x)^{2} \neq -f(x)\). Therefore, the function is not odd.

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