Question

     i.      Express  in the form , where a, b and c are constants.

   ii. Functions f and g are both defined for . It is given that  and     . Find .

  iii.       Find  and give the domain of .

 

Solution

     i.
 

We have the expression;

We use method of “completing square” to obtain the desired form.

We complete he square for the terms which involve .

We have the algebraic formula;

For the given case we can compare the given terms with the formula as below;

Therefore we can deduce that;

Hence we can write;

To complete the square we can add and subtract the deduced value of ;

   ii.
 

We are given that;

 We are required to find .

We have shown in (i) that  can be written as;

Therefore;

Since  is obtained by substituting  in . Therefore, for  and we are given , therefore;

  iii.
 

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 .

As demonstrated in (i), we can write the given function as;

Interchanging ‘x’ with ‘y’;

We are required to find domain   of .

We can find the domain    of  as follows.

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, if we can find the range    of  that will be domain   of .

So let’s find of range  of . 

To find the range  of  we need domain  of .

Let’s find domain  of .

Since  is a composite function its domain  is found as follows.

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

We have the function , therefore, domain  of  needs to be chosen such that the whole of the range  of  is included in the domain  of .

We are given that both functions  and  are defined for . Therefore, domain  of  function  is . Now we find the range of .

Finding range  of a function :

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

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

Therefore, range of g(x);

So the even for the given domain of , , the entire range of  is included in the given  domain of .

Hence, domain  of  is same as domain of  of  which is .

We can use this domain  of  to find range of  which will serve as domain of  .

Let’s find range of .

We have;

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  with ;

Hence, range of  is .

Therefore, domain of  is;