Cambridge Additional Mathematics

Functions (Chapter 2) 57

Example 18 Self Tutor

Consider f:x 7! 2 x+3.

a On the same axes, graphfand its inverse functionf¡^1.

b Findf¡^1 (x)using variable interchange.

c Check that (f±f¡^1 )(x)=(f¡^1 ±f)(x)=x.

a f(x)=2x+3passes through (0,3) and (2,7).

) f¡^1 (x) passes through (3,0)and (7,2).

b f is y=2x+3,

) f¡^1 is x=2y+3

) x¡3=2y

)

x¡ 3

2

=y

) f¡^1 (x)=

x¡ 3

2

c (f±f¡^1 )(x)

=f(f¡^1 (x))

=f

³

x¡ 3

2

́

=2

³

x¡ 3

2

́

+3

=x

and (f¡^1 ±f)(x)

=f¡^1 (f(x))

=f¡^1 (2x+3)

=(2x+3)¡^3

2

=^2 x

2

=x

Any function which has an inverse, and whose graph is symmetrical about the line y=x,isa

self-inverse function.

Iffis a self-inverse function then f¡^1 =f.

For example, the function f(x)=

1

x

, x 6 =0, is said to be

self-inverse, as f=f¡^1.

If includes point , ,

then includes point ,.

fab

ba

()

f-1 ()

y

x

y=x

y=%1__

x__

O

y

x

(0 3),¡

(3 0),¡

(7 2),¡

(2 7),¡

y = (x)¡¡f

y = f (x)¡¡-1

y=x¡¡

O

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_02\057CamAdd_02.cdr Thursday, 19 December 2013 2:45:58 PM BRIAN