Cambridge Additional Mathematics

Functions (Chapter 2) 55

However, not all functions have an inverse function.

For example, consider the function y=x^2. The inputs¡ 3 and 3 both produce an output of 9.

So, if we gave an inverse function the input 9 ,

how would it know whether the output should be

¡ 3 or 3? It cannot answer both, since the inverse

function would fail the vertical line test.

So, if a function has two inputs which produce the same output, then the function does not have an inverse

function.

For a function to have aninverse, the function must beone-one. It must pass the horizontal line test.

If y=f(x) has aninverse function, this new function:

² is denoted f¡^1 (x)

² is the reflection of y=f(x) in the line y=x

² satisfies (f±f¡^1 )(x)=x and (f¡^1 ±f)(x)=x.

The function y=x, defined as f:x 7 !x, is theidentity function.

f¡^1 isnotthe

reciprocal off.

f¡^1 (x) 6 =

1

f(x)

input

input

output

output

4

yx=2 +3

4

11

y = %%^x-3_

2____________

input

output

-3

y=x 2

9

input

output

3

9

y=x 2

input

inverse function output

9

-3or? 3

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_02\055CamAdd_02.cdr Tuesday, 26 November 2013 2:18:31 PM GR8GREG