Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  FebMar  (P19709/12)  Q#8
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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;
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