Cambridge Additional Mathematics

56 Functions (Chapter 2)

Example 17 Self Tutor

If y=f(x) has an inverse function, sketch y=f¡^1 (x), and state the domain and range of f(x)

and f¡^1 (x).

abc

a The function fails the horizontal line test, so it is not one-one. The function does not have an

inverse function.

b

c

FromExample 17, we can see that:

The domain off¡^1 is equal to the range off.

The range off¡^1 is equal to the domain off.

If(x,y)lies onf, then(y,x)lies onf¡^1. Reflecting the function in the line y=x has the algebraic

effect of interchangingxandy.

So, if the function is given as an equation, then we interchange the variables to find the equation of the

inverse function.

For example, iffis given by y=5x+2 thenf¡^1 is given by x=5y+2.

y = f (x)

y = f(x) y = x

is the reflection

of in the line.

-1

f(x) has domain fx:0 6 x 64 g

and range fy:¡ 16 y 62 g.

f¡^1 (x) has domain fx:¡ 16 x 62 g

and range fy:0 6 y 64 g.

f(x) has domain fx:x>¡ 2 g

and range fy:y> 0 g.

f¡^1 (x) has domain fx:x> 0 g

and range fy:y>¡ 2 g.

y

x

(0 -1), y = f(x)

(4 2),

O

y

x

y = f(x)

O

y

x

y = f(x)

-2 O

y

y = f(x) x

(2 4),

(4 2),

(-1 0),

(0 -1),

y=x

y = f-1(x)

O

y

x

y = f(x)

y=x

y = f-1(x)

-2 O

-2

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Additional Mathematics
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