Question

The function  is defined by  for , where a is a constant. The function  is  defined
 for .

     i.       Find the largest value of a for which the composite function can be formed.

For the case where ,

   ii.       solve the equation ,

  iii.       find the set of values of  which satisfy the inequality .

Solution

i.
 

We are given the two functions as;

 for

 for

It is evident that  has domain . Therefore, when we consider the composite function  the function  (or to be more specific the range of function ) becomes the domain of .  Hence, to meet the condition on domain of ;

We are given that  is the domain of , therefore; 

ii.
 

For the case where ,

 for 

 for

We are required to solve the equation;

It is evident that  has domain . Therefore, when we consider the composite function  the function  (or to be more specific the range of function ) becomes the domain of .

Hence, to meet the condition on domain of ;

As per requirement of  for  the values of  for the composite  function cannot exceed .

Hence only possibility of  for composite function ;

iii.
 

We are required to solve the inequality;

For the case where ,

 for 

 for

Therefore;

To find the set of values of x for which above inequality holds, we solve the following equation to  find critical values of ;

Now we  have two options;

These are critical values for the given inequality. 

From (i), we know that domain for the composite function ;

Hence,  is not possible. Therefore, only possible values of  are;