Question

The function f is such that  for . The function g is such that  for      , where a, b and q are constants. The function fg is such that  for .

i.       Find the values of a and b.

ii.       Find the greatest possible value of q.

It is now given that .

iii.       Find the range of fg.

iv.       Find an expression for  and state the domain of .

Solution

     i.
 

We are given that;

 for

We are also given that;

 for

 for

We can find  from given  and  functions;

We can compare the found and given  functions term by term;

Therefore

   ii.
 

We are given that;

 for

 for

We have found in (i) that;

Therefore;

 for

 for

We are required to find the greatest possible value of q.

We are given that  for . Therefore, domain of  can be written as;

Domain of :

By substituting extreme value of domain , we can find range of  as;

There are two approaches possible onwards in this question.

First is considering domain of .

When we deal with composite function  the  becomes domain of . As we  have seen above domain of  is , therefore;

Since we are given that  for , therefore we only consider   part and can  write it as;

Hence, greatest possible value of  is -2.

Second, is considering range of .

When we deal with composite function  the  becomes domain of . As we  have seen above range of  is , therefore;

Since we are given that  for , therefore we only consider   part and  can write it as;

Hence, greatest possible value of  is -2.

iii.
 

We are given that  for  and now we are given that .

Therefore;

 for

We can find range of  by substituting extreme value of domain ;

 

 

 

Therefore range of  can be expressed as;

  iv.
 

We have;

 for

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 .

As we have seen that , we consider only negative outcome of the radical;

Interchanging ‘x’ with ‘y’;

Now we are required to find the domain 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 

We have found in (iii) range of  as;

Therefore, we can write domain of  as;

Domain of =Range of

Hence for ;