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.