Question

The functions f and g are defined by

 for

 for .

     i.       Find and simplify an expression for  and state the range of .

 

   ii.       Find an expression for  and find the domain of .

Solution

     i.
 

We are given;

 for

 for .

We are required to find expression for ;

 

We are also required to find range of .

To find the range of of , we need domain of of .

For a composite function , the domain D of  must be chosen so that the whole of the  range of  is included in the domain of . The function , is then defined as , . 

Therefore, first we need to find range of  to compare it with domain of .

Let’s find range of .

Finding range of a function :

·       Substitute various values of  from given domain into the function to see what is happening to y. 

·       Make sure you look for minimum and maximum values of y by substituting extreme values of        from given domain.

We are given that;

 for

We can find the range of  by substituting some value(s) of domain of  into the function.

It is evident that range of function  is;

It is evident that domain of  must be chosen such that range of  is included in domain of  which is .

Therefore, entire range of  is included in the domain of .

Therefore, domain of  is also same as domain of .

Now let’s find range of .

Finding Range of a Function :

·       Substitute various values of  from given domain into the function to see what is happening to y.

·       Make sure you look for minimum and maximum values of y.

We are given that;

 

We can find the range of  by substituting some value(s) of domain of  into the function.

It is evident that range of function  is;

   ii.
 

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 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 (i) that range of  is;

Therefore;

Range of a function  Domain of

Hence;