Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#7

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Question

     i.       The diagram shows part of the curve  and part of the straight line  meeting at the point , where  and  are positive constants. Find the values of  and .

   ii.             The function f is defined for the domain  by

Express  in a similar way.

Solution


i.
 

It is evident from the diagram that point  is the point of intersection of the curve  and the straight line . Therefore, we are required to find the coordinates of point of intersection a curve and a line.

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate  coordinates of both lines i.e. equate equations of both the lines (or the line and the curve).

Equation of the line is;

Equation of the curve is;

Equating both equations;

Now we have two options;

Two values of x indicate that there are two intersection points.

However, it is evident from the diagram that point  is in first quadrant and hence its x-coordinate can be positive only. Therefore, we consider;

Corresponding value of y coordinate can be found by substituting value of x in any of the two equation i.e either equation of the line or equation of the curve.

We choose equation of line;

Hence, .


ii.
 

The function  is given piece-wise and hence we also find its inverse   piece-wise. 

We take the first part;

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 .

We are given that  and from (i) we can write as . Since  is always positive, we only consider;

Interchanging ‘x’ with ‘y’;

Since,  has its domain restricted to , the inverse  must also have its domain restricted.

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 

So to find domain of , it is easy to find range of .

Range of  can be found by substituting extreme values of its domain in its equation.

When ;

When ;

Hence range of  can be written as;

Therefore, domain of  can be written as;

Now we consider the second part of given function which is;

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’;

Since,  has its domain restricted to , the inverse  must also have its
domain restricted.

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 

So to find domain of , it is easy to find range of .

Range of  can be found by substituting extreme values of its domain in its equation.

When ;

Hence range of  can be written as;

Therefore, domain of  can be written as;

Now we can write the inverse of actual function, by combing its pieces according to found domains of each piece.

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