Chapter 1.8, Problem 40E

(a)

To determine

To find: The composite function $f\circ g$

Answer to Problem 40E

  $f\circ g=|-x-1|$

Explanation of Solution

Given information:

  $f(x)=|x-4|,g(x)=3-x$

Calculation.:

Find $f\circ g$

  $\begin{array}{l}f\circ g=f(g(x))\\ =f(3-x)\\ =|3-x-4|\\ =|-x-1|\end{array}$

Hence, $f\circ g=|-x-1|$

(b)

To determine

To find: The composite function $g\circ f$ and domains of each function and composite

functions.

Answer to Problem 40E

  $g\circ f=3-|x-4|$

Domains of

  $f(x),g\circ f,g(x)\&f\circ g\Rightarrow \text{all real numbers}$

Explanation of Solution

Given information:

  $f(x)=|x-4|,g(x)=3-x$

Calculation:

Find $g\circ f$

  $\begin{array}{l}g\circ f=g(f(x))\\ =g(|x-4|)\\ =3-|x-4|\end{array}$

Domains of

  $\begin{array}{l}f(x)\Rightarrow \text{all real numbers}\\ g(x)\Rightarrow \text{all real numbers}\\ f\circ g\Rightarrow \text{all real numbers}\\ g\circ f\Rightarrow \text{all real numbers}\end{array}$