Problem

For $f(x)=9 x-2$ and $g(x)=\frac{x+2}{9}$, find the following functions. a. $(f \circ g)(x) ;$ b. $(g \circ f)(x) ;$ c. $(f \circ g)(9) ;$ d. $(g \circ f)(9)$ a. $(f \circ g)(x)=$ (Simplify your answer.) b. $(g \circ f)(x)=$ (Simplify your answer.) c. $(f \circ g)(9)=$ d. $(g \circ f)(9)=$

Solution

Step 1 :First, we find the composition of the functions $f$ and $g$, denoted as $(f \circ g)(x)$, which means we substitute $g(x)$ into $f(x)$.

Step 2 :Substitute $g(x)$ into $f(x)$, we get $f(g(x)) = f\left(\frac{x+2}{9}\right) = 9\left(\frac{x+2}{9}\right) - 2$.

Step 3 :Simplify the above expression, we get $f(g(x)) = x + 2 - 2 = x$.

Step 4 :Next, we find the composition of the functions $g$ and $f$, denoted as $(g \circ f)(x)$, which means we substitute $f(x)$ into $g(x)$.

Step 5 :Substitute $f(x)$ into $g(x)$, we get $g(f(x)) = g(9x-2) = \frac{9x-2+2}{9}$.

Step 6 :Simplify the above expression, we get $g(f(x)) = x$.

Step 7 :Then, we find the value of $(f \circ g)(9)$, which means we substitute $9$ into $f(g(x))$.

Step 8 :Substitute $9$ into $f(g(x))$, we get $f(g(9)) = 9$.

Step 9 :Finally, we find the value of $(g \circ f)(9)$, which means we substitute $9$ into $g(f(x))$.

Step 10 :Substitute $9$ into $g(f(x))$, we get $g(f(9)) = 9$.

Step 11 :So, the final answers are $(f \circ g)(x) = \boxed{x}$, $(g \circ f)(x) = \boxed{x}$, $(f \circ g)(9) = \boxed{9}$, and $(g \circ f)(9) = \boxed{9}$.

From Solvely APP
Source: https://solvelyapp.com/problems/7925/

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