(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

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 .

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 .

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

Since the inverse function is a polynomial then its domain is all real numbers.

Domain of  is .

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

The range of the function is always greater than or equal to two.

Range of  is : .

(d)

Consider .

Differentiate the function with respect to .

Substitute  in above expression.

.

(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  and 

Range of  is : .

(d) .

(e) The graph is

.