Cambridge International Mathematics

2 Find the equation of the cubic function which:

a hasx-intercepts 1 and 3 , y-intercept 9 and passes through(¡ 1 ,8)

b touches thex-axis at 3 , hasy-intercept 18 and passes through(1,20).

3 The graph alongside has the form y=2x^3 +bx^2 +cx¡ 12.

Find the values ofbandc.

4 The graph alongside has the form y=¡x^3 +bx^2 +4x+d.

Find the values ofbandd.

Consider the mapping “add 5 ”:

¡ 2

0

1

4

“add 5 ”

3

5

6

9

Suppose we wanted to reverse this mapping, so we want to map 3 back to¡ 2 , 5 back to 0 , and so on.

To achieve this we would use the reverse

orinverseoperation of “add 5 ”, which is

“subtract 5 ”:

¡ 2

0

1

4

“subtract 5 ”

3

5

6

9

Theinverse function f¡^1 of a functionfis the function such that, for every value ofxthatfmaps

to f(x), f¡^1 mapsf(x)back tox.

From the above example, we can see that the inverse of f(x)=x+5 or “add 5 ”is f¡^1 (x)=x¡ 5

or “subtract 5 ”.

Consider the function f(x)=x^3 +4. The process performed by this function is to “cubex, then add 4 ”.

To find f¡^1 (x), we need to reverse this process. Using inverse operations, we “subtract 4 , then take the

cube root of the result”, and so f¡^1 (x)=^3

p

x¡ 4.

Unfortunately, it is not always so easy to reverse the process in a given function. However, there is an

algebraic method we can use to find the inverse function.

The inverse of y=f(x) can be found algebraically by interchangingxandy, and then makingythe

subject of the resulting formula. The newyis f¡^1 (x).

B INVERSE FUNCTIONS [3.9]

y

O x

-3

2

Further functions (Chapter 23) 473

O

y

x

2

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Y:\HAESE\IGCSE01\IG01_23\473IGCSE01_23.cdr Friday, 14 November 2008 12:16:55 PM PETER