Functions (Part 4)

function,AS Exam,CIE,completing the square,domain,range,inverse function,tangent,discriminant,line,curve

This item is taken from Cambridge International AS and A Level Mathematics (9709) Pure Mathematics 1 Paper 13 of May/June 2010.
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To deal with this item, let us summarize the details of the problem:
(a) Syllabus area: Functions
(b) Formula/Concept needed:
> completed square form of quadratics
> discriminant (b^2 - 4ac)
> domain and range of functions
> inverse of a function

Parts (i), (ii) and (iii) are done in the previous posts.

Now, let us look at the clue words/phrases in parts (iv) and (v).

For part (iv),

For part (v),

To restrict the domain, we must take note that the resulting function must be one-to-one. It means that it should pass the horizontal and vertical line tests.

Recall the graph of the given function.

The easiest way to restrict the parabola is to cut it along the vertex. In other words, cut the parabola into two (2) equal parts.

Separating the two parts, the domain will be restricted on both sides.

Since we would like that x ≥ A, it means that we are interested on the right side only. That is x ≥ 2.

Hence, the smallest value of of A is 2.

For (v), let us find the inverse of the function using the normal process.

function,AS Exam,CIE,completing the square,domain,range,inverse function,tangent line,line,curve

Since x ≥ 2, then we will take the positive square root.

Hence, the inverse of the function is

[End]

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