Functions
Neil Trivedi
Teacher
Function Notation
A function is a rule that takes an input, performs an operation on it, and produces an output. For each input, exactly one output is produced.
This idea can be illustrated using the example below.
Here, we take an input and put it into the function machine. The machine multiplies the input by and then adds The result is the output.
For a numerical example, suppose the input is First, we multiply by which gives Then, we add which gives Therefore, the output is
Now, suppose the input is First, we multiply by which gives Then, we add which gives Therefore, the output is
We write functions using function notation. For example, the function from before can be written as
where:
is the input.
is the rule applied to the input.
is the output.
To find the output of a function, we substitute the input into the function.
For example, to find if we substitute into the function.
Key Definitions:
What we put into a function is called the input.
What comes out is known as the output.
Example 1:
a) Write down what this function does.
The function multiplies the input by and then subtracts
b) Find
Single Step: is the input, which we substitute into the function to determine the output.
c) Find
Single Step: is the input, which substitute into the function to determine the output.
d) Find
Single Step: is the input, which we substitute into the function to determine the output.
e) Find
Single Step: is the input, which we substitute into the function to determine the output.
f) If work out the value of
Step 1: is the output. We form an equation using the function.
Step 2: Solve the equation for the input
Adding to both sides,
Dividing both sides by
Example 2:
Given that write down expressions for:
a)
Single Step: is the input, which we substitute into the function to determine the output.
b)
Single Step: is the input, which we substitute in place of
To simplify, we use the index rule which states that
c)
Single Step: Here, the input is The power of is on the outside so we apply it to the entire function, not just to
Using the index rule which states that and applying the power to both parts of the product,
Notice that Although they look similar, they produce different results.
d) Solve
Step 1: is the output. We form an equation using the function.
Step 2: Solve the equation for the input
Dividing both sides by
Square rooting both sides,
Inverse Functions
An inverse function reverses the effect of a function. If a function takes an input and produces an output the inverse function takes that output and returns it to the original input.
The inverse function is denoted by
Let’s revisit our function machine from the beginning.
The inverse function undoes the steps of the original function in reverse order, meaning we undo the last step first. So, to get from back to we first subtract and then divide by
Therefore, if then the function is
We can formalise the process of finding the inverse function using an algebraic method.
General Steps for Finding the Inverse Function
Suppose
Swap and in the equation. This gives an equation of the form
Rearrange the equation to make the subject.
Replace with
So, going back to our example, we have
Let
To find the inverse, we first swap and
Then, we rearrange the equation to make the subject. We start by subtracting from both sides.
Then, we divide both sides by
Finally, we replace with
Example 3:
If find
Step 1: Let then swap and in the equation.
Swapping and
Step 2: Rearrange the equation to make the subject.
First, we multiply both sides by
Adding to both sides,
Dividing both sides by
Step 3: Replace with
Example 4:
If find
Step 1: Let then swap and in the equation.
Swapping and
Step 2: Rearrange the equation to make the subject.
First, we multiply both sides by
Expanding the brackets on the left-hand side,
We need to collect the terms together on one side so next, we move the over to the right and the over to the right. We are essentially adding to both sides and subtracting from both sides.
On the left-hand side, we factorise out
Finally, dividing both sides by
Step 3: Replace with
It turns out that in this case. This means the function is equal to its own inverse and is called a self-inverse function.
Example 5:
If find
Step 1: Let then swap and in the equation.
Swapping and
Step 2: Rearrange the equation to make the subject.
This is a quadratic equation, in terms of so we cannot rearrange it directly. Although the quadratic can be factorised to become
this does not help us make the subject. Instead, we complete the square which rewrites our quadratic so the variable is in one place only. For more information on how to complete the square, see our “Completing the Square” note.
Completing the square on we get
Adding to both sides,
Square rooting both sides,
Subtracting from both sides,
Here, we have two possible solutions: one with a positive square root and one with a negative square root. However, for an inverse function, we only take one of these. So, we take the positive square root.
Step 3: Replace with
Note: The reason why we only take one of the square roots is studied in more detail in GCSE Further Maths and A-Level Maths. At GCSE maths, they accept answers that include also instead of the solution we provided, which was the positive root.
Composite Functions
Composite functions involve applying one function after another to an input . For example, means Here’s what happens:
1) The input is substituted into the function to get an output
2) Then, is substituted into the function to get the final output
With composite functions, the function inside is applied first.
This example is illustrated by the diagram below.
Example 6:
Consider the functions and such that
Find:
a)
Single Step: means so we apply first and then to determine the output.
is the input, which we substitute into the function
Next, we substitute into the function
Therefore,
b)
Single Step: means so we apply first and then to determine the output.
is the input, which we substitute into the function
Next, we substitute into the function
Therefore,
Notice how is different to Since we are applying the functions in different orders, the output changes.
c)
Single Step: means so we apply first and then again to determine the output.
is the input, which we substitute into the function
Next, we substitute into the function
Therefore,
Note: Another way we could have written is
d)
Single Step: means so we apply first and then to determine the output.
is the input so the output is just
Next, we substitute into the function
Therefore,
Example 7:
Consider the functions and such that
Solve
In this question, we are given the final output of and need to find the input We follow the usual process for composite functions to form an equation in terms of and then we solve for
Step 1: means so we apply first and then to determine the output.
is the input, which we substitute into the function
Next, we substitute into the function
Therefore,
Since
Step 2: Solve the equation for
First, we add to both sides.
Then, we square root both sides.
This gives us two equations to solve.
or
Adding 7 to both sides,
or
Dividing both sides by 2,
or
Therefore, the solutions of are or
Challenging Question