Question

for f(x)=9x - 4 and g(x)=$\frac{x + 4}{9}$, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(8); d. (g ∘ f)(8) a. (f ∘ g)(x)= (simplify your answer.)

Explanation:

Step1: Recall composition formula

$(f\circ g)(x)=f(g(x))$. Substitute $g(x)=\frac{x + 4}{9}$ into $f(x)$.
$f(g(x))=9(\frac{x + 4}{9})-4$

Step2: Simplify the expression

$9(\frac{x + 4}{9})-4=x + 4-4=x$

Step3: Recall composition formula for $(g\circ f)(x)$

$(g\circ f)(x)=g(f(x))$. Substitute $f(x)=9x - 4$ into $g(x)$.
$g(f(x))=\frac{(9x - 4)+4}{9}$

Step4: Simplify the expression for $(g\circ f)(x)$

$\frac{(9x - 4)+4}{9}=\frac{9x}{9}=x$

Step5: Find $(f\circ g)(8)$

Since $(f\circ g)(x)=x$, then $(f\circ g)(8)=8$

Step6: Find $(g\circ f)(8)$

Since $(g\circ f)(x)=x$, then $(g\circ f)(8)=8$

Answer:

a. $(f\circ g)(x)=x$
b. $(g\circ f)(x)=x$
c. $(f\circ g)(8)=8$
d. $(g\circ f)(8)=8$