# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#8

Question

The function  is such that  for , where a and b are constants.

i.       For the case where  and , find the possible values of  a and b.

ii.       For the case where  and , find an expression for  and give the domain of  .

Solution

i.

We are given that function  is represented as;

We are given that;

First we substitute  in the given expression of the function;

Equating it with given equation of ;

Secondly, we are given that;

Similarly, we substitute  in the given expression of the function;

Equating it with given equation of ;

From substitution of  in the given expression of the function and subsequently equating the  derived and given equations we have found that;

We can rearrange this to get;

Substituting in above equation we can;

Now we have two options;

Substituting these values of  in following equation we can find .

 For ; For ;

ii.

For the case where  and , the function  becomes;

for ;

We can write the given function as;

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

It is evident that to write the function in terms of  we need to write this quadratic equation in vertex  form first.

We have the expression;

Next 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 ;

Therefore, we can write the given function as;

We are given that function  for , therefore, only possibility for  is;

Interchanging ‘x’ with ‘y’;

Now 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

The given function, as demonstrated above, can be written as;

We are also given that given function is defined for

Hence, domain of the given function is

To find range of the function we can substitute extreme value(s) of the domain into the function.

Hence range of given function is;

Therefore;

Domain of  is;