Chapter 1, Problem 92RE

a.

To determine

Inverse of the function f.

Answer to Problem 92RE

Inverse of the function is $y=x^2-1$ .

Explanation of Solution

Given:

Function is given as;

  $f\left(x\right)=\sqrt{x+1}$

Let $y=f\left(x\right)$

Therefore,

  $y=\sqrt{x+1}$

Squaring both side.

  $\begin{array}{l}{y}^2=x+1\\ x=y^2-1\end{array}$

Now replacing x and y each other.

  $y=x^2-1$

Therefore inverse of the function is $y=x^2-1$ .

b.

To determine

The graph of the function and its inverse. .

Answer to Problem 92RE

  

Explanation of Solution

Given:

Function is given as;

  $f\left(x\right)=\sqrt{x+1}$

Graph of the function and its inverse is;

  

c.

To determine

Relationship between the graph of f and its inverse.

Answer to Problem 92RE

Both f and its inverse is the reflection of each other on line y=x

Explanation of Solution

Given:

Function is given as;

  $f\left(x\right)=\sqrt{x+1}$

Graph of the function and its inverse is;

  

Now the graph it is clear that both graph are reflection of each other on line y=x

d.

To determine

Domain and range of the function f and its inverse

Answer to Problem 92RE

Domain of the function is $\left(-1,\infty \right)$ and its range is $\left(0,\infty \right)$

Domain of $f^{-1}$ is $\left(0,\infty \right)$ and its range is $\left(-1,\infty \right)$

Explanation of Solution

Given:

Function is given as;

  $f\left(x\right)=\sqrt{x+1}$

Graph of the function and its inverse is;

  

From the graph it is clear that;

Domain of the function is $\left(-1,\infty \right)$ and its range is $\left(0,\infty \right)$

Domain of $f^{-1}$ is $\left(0,\infty \right)$ and its range is $\left(-1,\infty \right)$