1. Inverse functions

2. What is a function? 5 + 2 What is a function?
A function is a combination of one or more operations that convert an input into an output.
function
input
output
For example, y = x is a function that converts the input, x, into the output, y, 5 5
+ 2
+ 2
Teacher notes
This slide is a revision slide, see Functions and relations for a full explanation of functions.
and adding 2.
by multiplying the chosen input by 5
If we choose an input x = 6,
then y =
5 × = 32

3. Inputs and outputs We can use a graph to find the output from
a given input. In this case, the input is an x value and the output is a y value.
For example, x = 4 gives
y = 4
We can also use the same graph to find the input value for a given output.
For example, y = 6
is given by
This is the graph
of y = ½ x + 2
x = 8

4. Finding the input value
We can also find the input that gives
a certain output using algebraic methods.
In the function f(x) = 4x – 3, what value of x gives f(x) = 5?
substitute 5 into the equation:
5 = 4x – 3
5 + 3
rearrange:
x = 4
Teacher notes
Examples of situations where we might want to know the input for a given output might include:
how many units a seller must process in order to break even (you could stretch students by making this example non-linear by allowing discounts for bulk purchases, for example)
how much a person must earn before tax to in order to pay their bills
what test scores are needed to maintain a certain average.
Students should be able to use the above examples to form simple algebraic expressions. These expressions can be used to form functions and inverse functions with useful outputs. Try using the students’ own suggestions in the same way, or return to their suggestions at the end of the presentation as a plenary exercise.
For more information on the domain of functions, see the presentation Domain and range.
solve:
x = 2
Remember that x is the input, so it belongs to the domain of f.

5. This is called the inverse function and is written f–¹(x).
Inverse functions
If we are not given a value for f(x), we can rearrange the equation without substituting in a value:
f(x) + 3
f(x) = 4x – 3
becomes
x = 4
This equation gives the combination of operations that turns the output, f(x), back into the input, x.
Teacher notes
The inverse function is also often written f –1(y) in order to emphasize the idea that the inverse function takes the output, y, and gives the input, x, although there is no requirement that x be the input and y the output of the original function.
This is called the inverse function and is written f–¹(x).
f(x) + 3
x + 3
f –1(x) =
x = 4 4

6. Mapping diagrams
We can write the function f(x) = 4x – 3 using a mapping diagram showing the operations:
× 4
– 3 x 4x
4x – 3
To find the inverse of f(x) = 4x – 3, we can start with x and perform the inverse operations in reverse order.
÷ 4
+ 3
x + 3
Teacher notes
It may be useful to recap reverse operations.
x + 3 x 4
x + 3
And so for f(x) = 4x – 3,
f –1(x) = 4

7. Finding the inverse function
To find the inverse of f(x), rearrange to make x the subject.
becomes
old input, x
new output
becomes
old output, y
new input
What is the inverse of y = 3x + 2?
subtract 2 from both sides:
y – 2 = 3x
y – 2
divide both sides by 3:
= x 3
y – 2
write so that x is on the left:
x = 3
y – 2
f –1(y) = 3

8. Check your answer
If we apply f(x) to a number and then apply f –1(x) to the result, we should get back to the original number.
We can use this to check our results:
Let x = 2.
y – 2
f(x) = 3x + 2
f –1 (y) = 3
f (2) = (3 × 2) – 8
(–2) + 8
f –1 (–2) =
Image credit: © BruceParrott, Shutterstock.com 2012
= 6 – 8 3 6
f (2) = –2 = y = 3
f –1(–2) = 2 = x

9. Graphing inverse functions
Remember that the domain of f –1(x) is the range of f(x), and the range of f –1(x) is the domain of f(x).
The graph of y = f –1(x) is always a reflection of
y = f(x) about the line y = x.
y = ½ x + 2
y = x
y = 2x − 4