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2.3 Finding the inverse of a function

2.3 Finding the inverse of a function

You can represent a function in three di&erent ways.
 As a function machine  As an equation ‘ As a mapping

x × 2 + 3 y y = 2x + 3 x → 2x + 3

To work out the output values, substitute the input values into
the function. Input Output
In this function, the missing output values are: 5 ...
8 × 2 + 3 ...
5 × 2 + 3 = 13 and 8 × 2 + 3 = 19 ... 27
To work out the missing input value you take the output value and reverse the function.
−
So, in this function, the missing input value is: 27 3 = 12
2
When you reverse the function you are "nding the inverse function.

You can write an inverse function as a function machine, an equation or a mapping like this.

Function: x × 2 + 3 2x + 3
x
× 2
+ 3
2x + 3
Inverse: x – 3 ÷ 2 – 3 x
x – 3
2 ÷ 2 – 3 x
2
So the inverse function is y = x − 3 or x → x − 3 .
2 2

Worked example 2.3
Work out the inverse function for: a y = x + 5 b x → x −4 .
2

a
x + 5 x + 5 Draw the function machine.

x – 5 – 5 x Reverse the function machine.

y = x − 5 Write the inverse function as an equation.

b x
x ÷ 2 – 4 – 4 Draw the function machine.
2
2(x + 4) × 2 + 4 x Reverse the function machine.

x → 2(x + 4) Write the inverse function as a mapping.

) Exercise 2.3

1 Work out the inverse function for each equation.
a y = x + 9 b y = x − 1 c y = 3x d y = x
6

20 2 Sequences and functions