Question

a)Express in the form , where  and  are constants.

The function f is defined by  for .

b)Find an expression for and state the domain of .

The function is defined by  for .

c)For the case where k = −1, solve the equation .

d)State the largest value of possible for the composition  to be defined.

Solution

a)

We are given that;

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

 

b)

We are given that;

for

To find the inverse of a given function we need to write it in terms of rather than  in terms of .

We have found in (a) that the given function can be written as;

Since given function is defined for ,  that means  is not possible.

Hence;

Interchanging ‘x’ with ‘y’;

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 find range of , then we can find domain 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.

Therefore, domain of can be found from range of ;

We are given that function is defined for ; hence, domain of  is;

To find the range of , we substitute the least value of domain in the  function;

For ;

Hence range of  is;

Range of a function  Domain of

Hence, domain of ;

 

c)

The function f is for  can be written as.

The function  is defined by  for  when .

We are required to solve the equation;

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, range of  will be domain of .

Therefore, to find domain of we need to find range of .

We are given that  for  when .

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.

Hence, range of ;

It is evident that domain of  is;

Hence, we can conclude that only solution of the equation  is;

d)

The function f is for  can be written as.

The function  is defined by  for .

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 , .

Since for , for composite function to be valid, we  need to restrict range of to domain of .

We are given that domain of  is;

Therefore, range of  is;

Since for ;

For the range of will remain within domain of and hence, the  composite function will be valid.