# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#8

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Question

The function is defined by for , where a is a constant. The function is  defined for .

i.       Find the largest value of a for which the composite function can be formed.

For the case where ,

ii.       solve the equation ,

iii.       find the set of values of which satisfy the inequality .

Solution

i.

We are given the two functions as; for  for It is evident that has domain . Therefore, when we consider the composite function the function (or to be more specific the range of function ) becomes the domain of .  Hence, to meet the condition on domain of ;      We are given that is the domain of , therefore; ii.

For the case where , for  for We are required to solve the equation;          It is evident that has domain . Therefore, when we consider the composite function the function (or to be more specific the range of function ) becomes the domain of .

Hence, to meet the condition on domain of ; As per requirement of for the values of for the composite function cannot exceed .

Hence only possibility of for composite function ; iii.

We are required to solve the inequality; For the case where , for  for Therefore;      To find the set of values of x for which above inequality holds, we solve the following equation to  find critical values of ;     Now we  have two options;          These are critical values for the given inequality.

From (i), we know that domain for the composite function ; Hence, is not possible. Therefore, only possible values of are; 