For $f(x)=4 x-9$ and $g(x)=\frac{1}{4}(x+9)$, find $(f \circ g)(x)$ and $(g \circ f)(x)$. Then determine whether $(f \circ g)(x)=(g \circ f)(x)$.
What is $(f \circ g)(x) ?$
\[
(f \circ g)(x)=x
\]
What is $(g \circ f)(x)$ ?
\[
(g \circ f)(x)=
\]

Answer

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Answer

Final Answer: $\boxed{x}$

Steps

Step 1 ：Given functions are $f(x)=4 x-9$ and $g(x)=\frac{1}{4}(x+9)$.

Step 2 ：To find $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$: $f(g(x))=4(\frac{1}{4}(x+9))-9=x$.

Step 3 ：To find $(g \circ f)(x)$, substitute $f(x)$ into $g(x)$: $g(f(x))=\frac{1}{4}(4x-9+9)=x$.

Step 4 ：Comparing the two results, we find that $(f \circ g)(x)=(g \circ f)(x)$.

Step 5 ：Final Answer: $\boxed{x}$