Question

The function  is defined by  for .     

   i.       State the range of .

   ii.       Find the coordinates of the points at which the curve  intersects the coordinate                  axes. 

  iii.       Sketch the graph of .

  iv.       Obtain an expression for , stating both the domain and range of .

Solution

     i.
 

The given function is  for .

Domain of  is;

We find the values of  for extreme values of domain of ;

For ;

For ;

Therefore, range of ;

   ii.
 

We are required to find the coordinates of the points at which the curve  intersects the  coordinate axes ie we are looking for x and y intercepts of curve .

Let’s first find coordinates of x-intercept.

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Substitute ;

Hence coordinates of x-intercept are (0.253,0).

Let’s now find coordinates of y-intercept.

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Substitute ;

Hence coordinates of y-intercept are (0,1).

 

  iii.
 

We are required to sketch    for .

We can find the points of the graph as follows.

We also have coordinates of x and y intercepts of  as (0.253,0) and (0,1),  respectively.

Now we can sketch required graph from these points as shown below.

  iv.
 

We have;

We write it as;

To find the inverse of a given function  we need to write it in terms of  rather than in terms of .

Interchanging ‘x’ with ‘y’;

We are also required to find domain and range of .

Domain and range of a function  become range and domain, respectively, of its inverse function .

Domain of a function  Range of 

Range of a function  Domain of 

The given function is  for .

We have found in (i) that;

Domain of  is;

Range of ;

Therefore;

Domain of  is;

Range of ;