Cambridge Additional Mathematics

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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

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

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