(a)

The function is .

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 

.

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

(b)The function is .

Theorem 7:  If  is a one to one differentiable function with inverse function  and  then the inverse function is differentiable at  and .

Find .

Equate the function to .

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 .

Find the domain and range of .

The domain of a function is all values of , those make the function mathematically correct.

So, the domain of the inverse function is all real numbers.

Domain of  is .

Range set is the corresponding values of the function for different values of .

The range of the function is also all real numbers.

Range of  is  .

(d)

The inverse function is .

Differentiate the function with respect to .

Apply power rule of derivatives : .

Find  at .

. 

(e)

The graph of  and  is :

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

(b) .

(c)

The inverse function is ,

Domain of  is .

Range of  is  .

(d) .

(e)

The graph is : 

.