(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 :

.