Functions (Part 3)

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

Part (i) and part (ii) are done in the previous posts.

For (iii), it will be easier if we draw the graph of the quadratic function. The graph will show the minimum or maximum value of the range.

For the given function, the coefficient of x^2 is positive, then the graph has a minimum. It means that the parabola opens upward.

Let us first recall the vertex form of a quadratic:

From the previous items (posts), the completed square form of the given function is

In this function,the vertex is at (2, 6).

We can now draw the graph:

From the graph, we could see that the minimum value of y is 6. That is evident from the vertex of the parabola.

Hence,

*Note: In the actual exam, you are not required to draw the graph just to get the range, or even the domain. The graph is just a guide for you. However, if you are still struggling on how to find the domain and range without the graph, I would recommend you to at least sketch a part of the graph for you to visualise the function.

[End of part 3]
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