# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/11) | Q#9

Question

The function f is defined by for , .

i.        Find an expression for .

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

ii.       Find the value of a for which .

iii.       Find the possible values of a given that the equation has two equal roots.

Solution

i.

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 . Therefore;         Interchanging ‘x’ with ‘y’;  ii.

We have;  We can write these functions as;  Therefore, for ;  Hence, for ;  We are given that; Therefore;   iii.

We are given the equation; We have found in (i) that; Let’s find from the given function; 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 . Therefore;  Interchanging ‘x’ with ‘y’;  Therefore given equation can be written as;      We
are also given that equation has
two equal roots.

Therefore also has two equal roots.

Standard form of quadratic equation is; Expression for discriminant of a quadratic equation is; If ; Quadratic equation has two real roots.

If ; Quadratic equation has no real roots.

If ; Quadratic equation has one real root/two equal roots.

We can see that is a quadratic equation and has two equal roots. Therefore, in this case;          Now we have two options.      