Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  MayJun  (P19709/13)  Q#10
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Question
The oneone function f is defined by for , where c is a constant.
i. State the smallest possible value of c.
In parts (ii) and (iii) the value of c is 4.
ii. Find an expression for and state the domain of .
iii. Solve the equation , giving your answer in the form .
Solution
i.
Standard form of quadratic equation is;
The graph of quadratic equation is a parabola. If (‘a’ is positive) then parabola opens upwards and its vertex is the minimum point on the graph.
If (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the graph.
Vertex form of a quadratic equation is;
We have the function;
It is evident that given function is a quadratic equation in vertex form.
We need to find the smallest possible value of when so that is oneone function.
A oneone function has only one value of against one value of . A function is one one function if it passes horizontal line test i.e. horizontal line passes only through one point of function. However, if a function is not oneone, we can make it so by restricting its domain.
The given function is a parabola and parabola does not pass horizontal line test, so it is not a oneone function. However, we can make it so by restricting its domain around its line of symmetry.
The line of symmetry of a parabola is a vertical line passing through its vertex.
Therefore, we need coordinates of vertex of the given function.
Coordinates of the vertex are .Since this is a parabola opening upwards the vertex is the minimum point on the graph. Here ycoordinate of vertex represents least value of and xcoordinate of vertex represents corresponding value of .
For the given case, vertex is . Hence the vertical line passing through the vertex is . By restricting the domain of the given function to , we can make it oneone function.
Therefore, smallest possible value of is .
ii.
We are given that for ;
To find the inverse of a given function we need to write it in terms of rather than in terms of .
Since given function is defined for , that means is not possible.
Hence;
Interchanging ‘x’ with ‘y’;
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
Therefore, domain of can be found from range of ;
We are given that for ;
To find the range of , we substitute the least value of domain in the function;
Hence range of is;
Range of a function Domain of
Hence, domain of ;
iii.
We are required to solve the equation;
We are given that;
Since ;




We are given that for ;
Therefore, is not possible, hence, only possible value is;
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