Cambridge International Mathematics

Thehorizontal line testsays that ‘for a function to have an inverse function, no horizontal line can cut

it more than once.’

Example 4 Self Tutor

Find f¡^1 (x) for: a f(x)=8¡ 3 x b f(x)=

10

x+1

a By interchangingxandy, the inverse of

y=8¡ 3 x

is x=8¡ 3 y

) 3 y=8¡x

) y=

8 ¡x

3

) f¡^1 (x)=

8 ¡x

3

b By interchangingxandy, the inverse of

y=

10

x+1

is x=

10

y+1

) x(y+1)=10

) xy+x=10

) xy=10¡x

) y=

10 ¡x

x

) f¡^1 (x)=

10 ¡x

x

EXERCISE 23B

1 Find f¡^1 (x) for each of the following functions:

a f(x)=x¡ 7 b f(x)=3x+2 c f(x)=

3 ¡ 2 x

4

d f(x)=x^3 e f(x)=2x^3 +1 f f(x)=

4 x¡ 1

3

g f(x)=

p

x+1 h f(x)=

p

3 x¡ 5 i f(x)=

1

x¡ 2

2aFind the inverse function of: i f(x)=8¡x ii f(x)=

9

x

b What do you observe from your answers ina?

3aShow that the inverse of a linear function is also linear.

b What is the relationship between the gradient of a linear function and the gradient of its inverse?

c Explain why the following statement is true:

d

iiiy

O x

-1 ()2 -2,

“If(a,b) lies on y=f(x)=mx+c, then (b,a) lies on y=f¡^1 (x).”

Find the inverse function of:

y

x

O

2

()6 ¡1, ¦()x

¦()x

474 Further functions (Chapter 23)

IGCSE01

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Y:\HAESE\IGCSE01\IG01_23\474IGCSE01_23.CDR Monday, 27 October 2008 2:18:39 PM PETER