Chapter 0.2, Problem 52E

a.

To determine

To find : $f\left(g\left(x\right)\right)$ ,where $f\left(x\right)=x+1,g\left(x\right)=x-1$ .

Answer to Problem 52E

The value of $f\left(g\left(x\right)\right)=x.$

Explanation of Solution

Given information:

The given function are $f\left(x\right)=x+1,g\left(x\right)=x-1.$

Concept used:

The composition $f\text{of}\text{g}\text{of}\text{x}$ is the composition $f$ with $g$ .The value of $f^{o}g\left(x\right)=f\left(g\left(x\right)\right)$ .

Calculations:

Here $g\left(x\right)=x-1$ .

Replace $x$ by $g\left(x\right)=x-1$ in $f\left(x\right)$ to get $f\left(g\left(x\right)\right)$ .

  $\begin{array}{c}g\left(x\right)=x-1\\ f\left(g\left(x\right)\right)=f\left(x-1\right)\\ =x-1+1\\ =x\end{array}$

Thus the value of $f\left(g\left(x\right)\right)=x$ .

b.

To determine

To find : $g\left(f\left(x\right)\right)$ ,where $f\left(x\right)=x+1,g\left(x\right)=x-1.$ .

Answer to Problem 52E

The value of $g\left(f\left(x\right)\right)=x.$

Explanation of Solution

Given information:

The given function are $f\left(x\right)=x+1,g\left(x\right)=x-1.$

Concept used:

The composition $f\text{of}\text{g}\text{of}\text{x}$ is the composition $f$ with $g$ .The value of $f^{o}g\left(x\right)=f\left(g\left(x\right)\right)$ .

Calculations:

Here $g\left(x\right)=x-1$ .

Replace $x$ by $f\left(x\right)=x+1$ in $f\left(x\right)$ to get $g\left(f\left(x\right)\right)$ .

  $\begin{array}{c}g\left(x\right)=x-1\\ g\left(f\left(x\right)\right)=g\left(x+1\right)\\ =\left(x+1\right)-1.\\ =x\end{array}$

Thus the value of $g\left(f\left(x\right)\right)=x$

c.

To determine

To find : $f\left(g\left(0\right)\right)$ ,where $f\left(x\right)=x+1,g\left(x\right)=x-1.$ .

Answer to Problem 52E

The value of $f\left(g\left(0\right)\right)=0.$

Explanation of Solution

Given information:

The given function are $f\left(x\right)=x+1,g\left(x\right)=x-1.$

Concept used:

The composition $f\text{of}\text{g}\text{of}\text{x}$ is the composition $f$ with $g$ .The value of $f^{o}g\left(x\right)=f\left(g\left(x\right)\right)$ .

Calculations:

Here $g\left(x\right)=x-1$

Replace $x$ by $g\left(x\right)=x-1$ in $f\left(x\right)$ to get $f\left(g\left(x\right)\right)$ .

Now substitute $x=0$ into $f\left(g\left(x\right)\right)$ .

  $\begin{array}{c}f\left(g\left(x\right)\right)=x\\ f\left(g\left(0\right)\right)=0\\ \end{array}$

Thus the value of $f\left(g\left(0\right)\right)=0.$

d.

To determine

To find : $g\left(f\left(0\right)\right)$ ,where $f\left(x\right)=x+1,g\left(x\right)=x-1$ .

Answer to Problem 52E

The value of $g\left(f\left(0\right)\right)=0$

Explanation of Solution

Given information:

The given function are $f\left(x\right)=x+1,g\left(x\right)=x-1.$

Concept used:

The composition $f\text{of}\text{g}\text{of}\text{x}$ is the composition $f$ with $g$ .The value of $f^{o}g\left(x\right)=f\left(g\left(x\right)\right)$ .

Calculations:

Here $g\left(x\right)=x-1$ .

Replace $x$ by $f\left(x\right)=x+1$ in $f\left(x\right)$ to get $g\left(f\left(x\right)\right)$ .

  $\begin{array}{c}g\left(x\right)=x-1\\ g\left(f\left(x\right)\right)=g\left(x+1\right)\\ =x+1-1\\ =x.\end{array}$

Thus the value of $g\left(f\left(x\right)\right)=x$

Now substitute $x=0$ into $g\left(f\left(x\right)\right)$ .

  $\begin{array}{c}g\left(f\left(x\right)\right)=x\\ g\left(f\left(0\right)\right)=0\\ \end{array}$

Thus the value of $g\left(f\left(0\right)\right)=0$

e.

To determine

To find : $f\left(f\left(x\right)\right)$ ,where $f\left(x\right)=x+1,g\left(x\right)=x-1.$ .

Answer to Problem 52E

The value of $f\left(f\left(x\right)\right)=x+2.$

Explanation of Solution

Given information:

The given function are $f\left(x\right)=x+1,g\left(x\right)=x-1.$

Concept used:

The composition $f\text{of}\text{g}\text{of}\text{x}$ is the composition $f$ with $g$ .The value of $f^{o}g\left(x\right)=f\left(g\left(x\right)\right)$ .

Calculations:

Here $f\left(x\right)=x+1$ .

Replace $x$ by $f\left(x\right)=x-1$ in $f\left(x\right)$ to get $f\left(f\left(x\right)\right)$ .

  $\begin{array}{c}f\left(x\right)=x+1\\ f\left(f\left(x\right)\right)=f\left(x+1\right)\\ =x+1+1\\ =x+2\end{array}$

Thus the value of $f\left(f\left(x\right)\right)=x+2.$

f.

To determine

To find : $g\left(g\left(-2\right)\right)$ ,where $f\left(x\right)=x+1,g\left(x\right)=x-1$ .

Answer to Problem 52E

The value of $g\left(g\left(-2\right)\right)=-4.$

Explanation of Solution

Given information:

The given function are $f\left(x\right)=x+1,g\left(x\right)=x-1.$

Concept used:

The composition $f\text{of}\text{g}\text{of}\text{x}$ is the composition $f$ with $g$ .The value of $f^{o}g\left(x\right)=f\left(g\left(x\right)\right)$ .

Calculations:

Here $g\left(x\right)=x-1$ .

Replace $x$ by $g\left(x\right)=x-1$ in $f\left(x\right)$ to get $g\left(g\left(x\right)\right)$ .

  $\begin{array}{c}g\left(x\right)=x-1\\ g\left(g\left(x\right)\right)=g\left(x-1\right)\\ =x-1-1\\ =x-2.\end{array}$

Now substitute $x=-2$ into $g\left(g\left(x\right)\right)$ .

  $\begin{array}{c}g\left(g\left(x\right)\right)=x-2\\ g\left(g\left(-2\right)\right)=-2-2\\ =-4\end{array}$

Thus the value of $g\left(g\left(-2\right)\right)=-4.$