Exercise 1.3.69

Question

Suppose $g$ is an even function and let $h=f \circ g$. Is $h$ always an even function?

Khagan Gulusoy · Accepted Answer

To prove that $f \circ g=h$ is an even function, we should prove the main property of even functions:
\begin{equation}
\forall x \in X: \quad f \circ g(x)=f \circ g(-x),
\end{equation}
where $X$ is the domain of the function $f \circ g$. Since $g$ is an even function, we have $\forall x \in D: g(x)=g(-x)$, where $D$ is the domain of $g$. Therefore, $(1)$ holds, and $f \circ g=h$ is an even function. $\square$

Exercise 1.3.69

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Exercise 1.3.69

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