Chapter 4.2, Problem 2E

a.

To determine

To find: The composition of the functions. $$

Answer to Problem 2E

  $\begin{array}{l}f\circ g=17-14x.\\ g\circ f=-14x-29.\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=2x+5\\ g(x)=6-7x\end{array}$

Calculation:

From the given information,

  $\begin{array}{l}f\circ g=f(g(x))=2(6-7x)+5\\ f\circ g=12-14x+5\\ f\circ g=17-14x\end{array}$

And

  $\begin{array}{l}g\circ f=g(f(x))=6-7(2x+5)\\ g\circ f=6-14x-35\\ g\circ f=-14x-29\end{array}$

b.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=4x^2+8x+3.\\ g\circ f=2x^2+4x+1.\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=x^2+2x\\ g(x)=2x+1\end{array}$

Calculation:

From the given information,

  $\begin{array}{l}$ f\circ g=f(g(x))={(2x+1)}^2+2(2x+1)$f\circ g=4x^2+4x+1+4x+2\\ f\circ g=4x^2+8x+3\end{array}$

And

  $\begin{array}{l}g\circ f=g(f(x))=2(x^2+2x)+1\\ g\circ f=2x^2+4x+1\end{array}$

c.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=\sin (2x^2+1).\\ g\circ f=2{\sin }^{2}x+1.\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=\sin x\\ g(x)=2x^2+1\end{array}$

Calculation:

From the given information,

  $\begin{array}{l}f\circ g=f(g(x))=\sin (2x^2+1)\\ f\circ g=\sin (2x^2+1)\end{array}$

And

  $\begin{array}{l}f(x)=\sin x\\ $ g\circ f=g(f(x))=2{(\sin x)}^2+1$g\circ f=2{\sin }^{2}x+1\end{array}$

d.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=\tan (\sin x).\\ g\circ f=\sin (\tan x).\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=\tan x\\ g(x)=\sin x\end{array}$

Calculation:

From the given information,

  $\begin{array}{l}f\circ g=f(g(x))=\tan (\sin x)\\ f\circ g=\tan (\sin x)\end{array}$

  $\begin{array}{l}g\circ f=g(f(x))=\sin (\tan x)\\ g\circ f=\sin (\tan x)\end{array}$

e.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=\{(k,r),(t,r),(s,l)\}.\\ g\circ f=\varnothing .\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=\{(t,r),(s,r),(k,l)\}\\ g(x)=\{(k,s),(t,s),(s,k)\}\end{array}$

Calculation:

From the given information,

  $\begin{array}{l}f\circ g=f(g(x))\\ f\circ g=\{(k,r),(t,r),(s,l)\}\end{array}$

  $g\circ f=\varnothing$

f.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=\{(1,2),(2,5),(4,5),(5,2)\}.\\ g\circ f=\{(1,7),(3,4),(4,3),(5,3)\}.\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=\{(1,3),(2,6),(3,5),(4,2),(5,2)\}\\ g(x)=\{(1,5),(2,3),(3,7),(4,3),(5,4)\}\end{array}$

Calculation:

From the given information,

  $\begin{array}{l}f\circ g=f(g(x))\\ f\circ g=\{(1,2),(2,5),(4,5),(5,2)\}\\ And\\ g\circ f=g(f(x))\\ g\circ f=\{(1,7),(3,4),(4,3),(5,3)\}\end{array}$

g.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=\frac{x^2+2}{x^2+3}.\\ g\circ f=\frac{2x^2+6x+5}{x^2+4x+4}.\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=\frac{x+1}{x+2}\\ g(x)=x^2+1\end{array}$

Calculation:

From the given information,

  $\begin{array}{l}f\circ g=f(g(x))=\frac{x^2+1+1}{x^2+1+2}\\ f\circ g=\frac{x^2+2}{x^2+3}\\ And\\ $ g\circ f=g(f(x))={(\frac{x+1}{x+2})}^2+1$g\circ f=\frac{x^2+2x+1}{x^2+4x+4}+1\\ g\circ f=\frac{x^2+2x+1+x^2+4x+4}{x^2+4x+4}\\ g\circ f=\frac{2x^2+6x+5}{x^2+4x+4}\end{array}$

h.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=3\left|x\right|+2.\\ g\circ f=|3x+2|.\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=3x+2\\ g(x)=\left|x\right|\end{array}$

Calculation:

From the given information,

  $\begin{array}{l}f\circ g=f(g(x))=3\left|x\right|+2\\ f\circ g=3\left|x\right|+2\\ And\\ g\circ f=g(f(x))=|3x+2|\\ g\circ f=|3x+2|\end{array}$

i.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=2x+1,x\le -1\\ g\circ f=2x+2,x\le -1\\ f\circ g=-x+1,-1<x\le 0\\ g\circ f=-x-1,-1<x\le 0\\ f\circ g=-2x,x>0\\ g\circ f=-2x,x>0.\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=x+1,if x\le 0\\ f(x)=2x,if x>0\end{array}$

  $\begin{array}{l}g(x)=2x,if x\le -1\\ g(x)=-x,if x>-1\end{array}$

Calculation:

From the given information,

Case 1: $x\le -1$

  $\begin{array}{l}f\circ g=f(g(x))=2x+1\\ g\circ f=g(f(x))=2(x+1)=2x+2\end{array}$

Case 2: $-1<x\le 0$

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

Case 3: $x>0$

  $\begin{array}{l}f\circ g=f(g(x))=-2x\\ g\circ f=g(f(x))=-2x\end{array}$

j.

To determine

To find: The composition of the functions.

Answer to Problem 2E

  $\begin{array}{l}f\circ g=17-4x,x\le 2\\ g\circ f=1-4x,x\le 2\\ f\circ g=2x+5,2<x<3\\ g\circ f=2x+4,2<x<3\\ $ f\circ g={(7-2x)}^2,x\ge 3$g\circ f=x^2+1,x\ge 3\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=2x+3,if x<3\\ f(x)=x^2,if x\ge 3\end{array}$

  $\begin{array}{l}g(x)=7-2x,if x\le 2\\ g(x)=x+1,if x>2\end{array}$

Calculation:

From the given information,

Case 1: $x\le 2$

  $\begin{array}{l}f\circ g=f(g(x))=2(7-2x)+3\\ f\circ g=17-4x\\ g\circ f=g(f(x))=7-2(2x+3)\\ g\circ f=1-4x\end{array}$

Case 2: $2<x<3$

  $\begin{array}{l}f\circ g=f(g(x))=2(x+1)+3=2x+5\\ g\circ f=g(f(x))=-(x+1)=2x+3+1=2x+4\end{array}$

Case 3: $x\ge 3$

  $\begin{array}{l}$ f\circ g=f(g(x))={(7-2x)}^2$g\circ f=g(f(x))=x^2+1\end{array}$