Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2017  FebMar  (P19709/12)  Q#8
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Question
The functions f and g are defined for by
i. Show that and obtain an unsimplified expression for .
ii. Find an expression for and determine the domain of .
iii. Solve the equation .
Solution
i.
We are given the functions;
We can rewrite these as;
First we find .
Next we find .
ii.
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 .
Interchanging ‘x’ with ‘y’;
We are required to find the domain of .
The set of numbers for which function is defined is called domain of the function.
We recognize that is a composite function.
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 , .
There is relationship between range and domain of inverse functions.
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, we can find range of which will be domain of .
The set of values a function can take against its domain is called range of the function.
Hence, we are looking for range of a composite function .
To find the range of first we need to find the domain of .
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, we need range of .
We know that function is defined for . Hence, we can find its range.
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 , by substituting values of allowed by domain of , we get;
We are given that function is defined for . It is evident that whole range of is included in the domain of .
Therefore, range of can be found from domain of directly.
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.
We have found that;
Therefore, range of , by substituting values of allowed by domain of , we get;
Since;
Range of a function Domain of
Therefore, domain of is;
iii.
We are required to solve the equation;
We have found in (i) that;
Therefore;
Hence;
Now we have two options.



