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 $(f \circ g)(x)=(g \circ f)(x)$.
Solution
Step 1 ：We are given two functions, \(f(x) = 4x - 9\) and \(g(x) = \frac{1}{4}(x + 9)\).
Step 2 ：We are asked to find the compositions of these functions, \((f \circ g)(x)\) and \((g \circ f)(x)\), and then determine whether they are equal.
Step 3 ：First, let's find \((f \circ g)(x)\), which means we apply function \(f\) to the results of function \(g\).
Step 4 ：Substitute \(g(x)\) into \(f(x)\), we get \(f(g(x)) = 4(\frac{1}{4}(x + 9)) - 9 = x\).
Step 5 ：Next, let's find \((g \circ f)(x)\), which means we apply function \(g\) to the results of function \(f\).
Step 6 ：Substitute \(f(x)\) into \(g(x)\), we get \(g(f(x)) = \frac{1}{4}(4x - 9 + 9) = x\).
Step 7 ：Finally, we compare \((f \circ g)(x)\) and \((g \circ f)(x)\). Both simplify to \(x\), so they are indeed equal.
Step 8 ：Final Answer: \((f \circ g)(x) = x\), \((g \circ f)(x) = x\), and \((f \circ g)(x) = (g \circ f)(x)\), so the answer is \(\boxed{True}\).
