# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#7

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Question

The functions f and g are defined by for  for .

i.
Find the range of f and the range of g.

ii.       Find an expression for fg(x), giving your answer in the form , where a, b and c are  integers.

iii.      Find an expression for , giving your answer in the same form as for part (ii).

Solution

i.

The given function is for .

Therefore, domain of is; For range of we substitute extreme value(s) of the domain in the function to get the extreme values of the range of the function. It is evident that variable quantity in the given function is . It can assume any value based on . For the minimum possible value of the will be minimum and consequently will assume the maximum possible value.

Therefore, if ; Since, , .

Similarly, for the maximum possible value of the will be maximum and consequently will assume the maximum possible value which will be always greater than ZERO.

Hence, the rage of for is; The given function is for .

Therefore, domain of is; For range of we substitute extreme value(s) of the domain in the function to get the extreme values of the range of the function. It is evident that variable quantity in the given function is . It can assume any value based on .

For the minimum possible value of the will be maximum and consequently will assume the maximum possible value.

Therefore, if   However, since ; Similarly, for the maximum possible value of the will be minimum and consequently will assume the minimum possible value. Hence, the rage of for is; ii.

We are given; for  for We are required to find the rage of function .

First, we find expression for ;      iii.

Next, we find .

We have found above that; 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’;   