Question

The functions f and g are defined for  by

i.       Show that  and obtain an unsimplified expression for .

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

iii.       Solve the equation .

Solution

     i.
 

We are given the functions;

We can rewrite these as;

First we find .

Next we find .

   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 required to find the domain of .

The set of numbers  for which function  is defined is called domain of the function.

We recognize that  is a composite function.

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 , .

There is relationship between range and domain of inverse functions.

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 

Therefore, we can find range of  which will be domain of .

The set of values a function  can take against its domain is called range of the function.

Hence, we are looking for range of a composite function .

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

We know that function  is defined for . Hence, we can find its range.

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.

Therefore, range of , by substituting values of  allowed by domain of , we get;

We are given that function  is defined for . It is evident that whole range of  is included in the domain of .

Therefore, range of  can be found from domain of  directly.

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 have found that;

Therefore, range of , by substituting values of  allowed by domain of , we get;

Since;

Range of a function  Domain of

Therefore, domain of  is;

  iii.
 

We are required to solve the equation;

We have found in (i) that;

Therefore;

Hence;

Now we have two options.