Q. 43 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 29 – 44, for the given functions f and g, find: (a) $f\circ g$ (b) $g\circ f$ (c) $f\circ f$

 (d) $g\circ g$

State the domain of each composite function.

$f\left(x\right)=\frac{x-5}{x+1},g\left(x\right)=\frac{x+2}{x-3}$

Step-by-Step Solution

Answer

(a) $f\circ g=\frac{-4x-13}{-2x+11}$and domain is $R-\left\{\frac{1}{2}\right\}$

(b) $g\circ f=\frac{3x-3}{-2x-8}$and domain is $R-\left\{-4\right\}$

(c) $f\circ f=\frac{-x-5}{x-2}$and domain is $R-\left\{2\right\}$

(d) $g\circ g=\frac{3x-4}{-2x+11}$and domain is $R-\left\{\frac{11}{2}\right\}$

Step 1. Given Information

The given functions are $f\left(x\right)=\frac{x-5}{x+1},g\left(x\right)=\frac{x+2}{x-3}$

Part (a) Step 1. Finding f ∘ g

$$\begin{gathered} f\circ g=f\left(g\left(x\right)\right)=f\left(\frac{x+2}{x-3}\right) \\ f\circ g=\frac{{\displaystyle \frac{x+2}{x-3}}-5}{\frac{x+2}{x-3}-1} \\ f\circ g=\frac{{\displaystyle \frac{x+2-5x+15}{x-3}}}{{\displaystyle \frac{x+2+x-3}{x-3}}} \\ f\circ g=\frac{-4x+17}{2x-1} \end{gathered}$$

Part (a) Step 2. Finding domain of f ∘ g

Put $$\begin{gathered} 2x-1=0 \\ 2x=1 \\ x=\frac{1}{2} \end{gathered}$$

The domain is the set of all real numbers except $\frac{1}{2}.$

Part (b) Step 1. Finding g ∘ f

$$\begin{gathered} g\circ f=g\left(f\left(x\right)\right)=g\left(\frac{x-5}{x+1}\right) \\ g\circ f=\frac{{\displaystyle \frac{x-5}{x+1}}+2}{\frac{x-5}{x+1}-3} \\ g\circ f=\frac{{\displaystyle \frac{x-5+2x+2}{x+1}}}{{\displaystyle \frac{x-5-3x-3}{x+1}}} \\ g\circ f=\frac{3x-3}{-2x-8} \end{gathered}$$

Part (b) Step 2. Finding domain of g ∘ f

Put $$\begin{gathered} -2x-8=0 \\ -2x=8 \\ x=-4 \end{gathered}$$

The domain is the set of all real numbers except $-2.$

Part (c) Step 1. Finding f ∘ f

$$\begin{gathered} f\circ f=f\left(f\left(x\right)\right)=f\left(\frac{x-5}{x+1}\right) \\ f\circ f=\frac{{\displaystyle \frac{x-5}{x+1}}-5}{{\displaystyle \frac{x-5}{x+1}}+1}=\frac{x-5-5x-5}{x-5+x+1} \\ f\circ f=\frac{-4x-10}{2x-4}=\frac{-2\left(x+5\right)}{2\left(x-2\right)}=\frac{-x-5}{x-2} \end{gathered}$$

Part (c) Step 2. Finding domain

Put $$\begin{gathered} x-2=0 \\ x=2 \end{gathered}$$

The domain is the set of all real number except $2.$

Part (d) Step 2. Finding g ∘ g

$$\begin{gathered} g\circ g=g\left(g\left(x\right)\right)=g\left(\frac{x+2}{x-3}\right) \\ g\circ g=\frac{{\displaystyle \frac{x+2}{x-3}}+2}{{\displaystyle \frac{x+2}{x-3}}-3}=\frac{x+2+2x-6}{x+2-3x+9} \\ g\circ g=\frac{3x-4}{-2x+11} \end{gathered}$$

Part (d) Step 2. Finding domain of g ∘ g

Put $$\begin{gathered} -2x+11=0 \\ -2x=-11 \\ x=\frac{11}{2} \end{gathered}$$

The domain is the set of all real numbers except $\frac{11}{2}.$