(a)

The function is ,  and .

A function  is said to be one to one if any two elements in the domain are correspond to two different elements in the range.

If  and  are two different inputs of a function , then  is said to be one to one provided  .

If  then  for .

Multiply on each side by .

.

Add  to each side.

.

.

Therefore, the function   is said to be one-to-one function.

(b)

The function is .

Theorem 7:

If  is a oneto one differentiable function with inverse function  and  then the inverse function is differentiable at  and .

Find .

Equate the function to .

 since  is not in the interval .

Therefore  then .

.

Differentiate the function with respect to .

.

Power rule of derivatives : .

.

.

(c)

The function is .

Let .

To find the inverse of , replace  with  and  with .

.

Solve for .

The inverse of the function  is .

From the inverse function definition,

Domain of  is range of  and the range of  is domain of .

Domain of  is .

Range of  is  .

Therefore,

Domain of  is .

Range of  is  .

(d)

The inverse function is .

Differentiate on each side with respect to .

Power rule of derivatives : .

Sunbstitute  in above expression.

.

Sunbstitute  in above expression.

.

(e)

The graph of  and  is :

(a) The function   is one-to-one function.

(b) .

(c)

The inverse function is ,

Domain of  is .

Range of  is  .

(d) 

.

(e)

The graph is :

.