Q. 37 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 29 – 44, for the given functions f and g, find:

(a) $f o g$

 (b) $g o f$

(c) $f o f$

 (d)  $g o g$

State the domain of each composite function.  

$$\begin{gathered} f\left(x\right)=\frac{x}{x-1} \\ g\left(x\right)=-\frac{4}{x} \end{gathered}$$

Step-by-Step Solution

Answer

a)$f\circ g=\frac{4}{4+x}$ and its domain is:$\left\{x\right|x\ne 0,x\ne -4\}$

b) $g\circ f=-\frac{4(x-1)}{x}$, and its domain:$\left\{x|x\ne 1,x\ne 0\right\}$

c)$f\circ f=x$  and its domain is:$\left\{x|x\ne 1\right\}$.

d)$g\circ g=x$ and its domain is:$\left\{x|x\ne 0\right\}$

Step 1. Given information:

The functions are:

$$\begin{gathered} f\left(x\right)=\frac{x}{x-1} \\ g\left(x\right)=-\frac{4}{x} \end{gathered}$$

The domain of $f\left(x\right)$ is: $\left\{x\right|x\ne 1\}$

The domain of $g\left(x\right)$ is:$\left\{x\right|x\ne 0\}$

Part (a) Step 1. To find f   ∘ g and its domain:

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

The domain of $f\circ g$ is: $\left\{x\right| x\ne -4, x\ne 0\}$

Part (b) Step 1. To find g ∘   f and its domain:

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

The domain of $g\circ f$ is: $\left\{x|x\ne 0,x\ne 1\right\}$.

Part (c) Step 1. To find f ∘ f and its domain:

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

Its domain is:$\left\{x|x\ne 1\right\}$

Part (c) Step 1. To find g ∘ g and its domain:

$$\begin{gathered} g\circ g=-\frac{4}{-{\displaystyle \frac{4}{x}}} \\ =x \end{gathered}$$

its domain is:$\left\{x|x\ne 0\right\}$