Functions (AQA GCSE Maths): Revision Notes
Functions
What are functions?
A function is a mathematical relationship that takes an input value, processes it using a specific rule, and produces an output value. Think of it like a machine that follows the same set of instructions every time you feed it a number.
Functions can be written in two different ways:
- Function notation:
- Mapping notation:
Both expressions mean exactly the same thing: "the function f takes a value for x, multiplies it by 5, and adds 2". Don't let the mathematical appearance intimidate you - functions are essentially equations with y replaced by f(x).
Evaluating functions
Evaluating a function means finding the output when you substitute a specific input value. This process is straightforward - you simply replace the variable with the given number and calculate the result.
Worked Example: Function Evaluation
If , find and
For :
- Substitute :
- Calculate:
For :
- Substitute :
- Calculate:
The key is to be careful with negative numbers and follow the order of operations correctly.
Combining functions
Sometimes you'll encounter questions involving two functions that need to be combined into a single function, creating what's called a composite function. This process involves applying one function to the output of another.
When you see notation like , this means "apply function g first, then apply function f to the result". The order is crucial - is generally not the same as .
Method for finding composite functions:
- Rewrite as
- Replace with its actual expression
- Substitute this expression into function f
Worked Example: Composite Functions
If and , find and
For :
- Substitute into f:
For :
- Substitute into g:
Notice how and are completely different functions, demonstrating why order matters in function composition.
Inverse functions
An inverse function reverses the operation of the original function. If transforms input into output , then the inverse function transforms back into .
The method for finding inverse functions follows three clear steps:
Method for finding inverse functions:
- Write out the equation
- Rearrange the equation to make the subject
- Replace with
Worked Example: Finding an Inverse Function
If , find
Step 1: Write
Step 2: Rearrange to make the subject
Step 3: Replace with
You can verify your answer by checking that the inverse function truly reverses the original. For instance, if , then should equal 9.
Practice and application
Understanding functions requires practice with different types of problems. You might encounter questions asking you to:
- Evaluate functions at specific points
- Find composite functions
- Calculate inverse functions
- Work with function notation and operations
The key to success with functions is recognising the pattern in each type of problem and applying the appropriate method systematically.
Key Points to Remember:
- Function evaluation: Simply substitute the input value for x and calculate the result
- Composite functions: means "do g first, then f" - order matters and
- Inverse functions: Use the three-step method - write , rearrange for , then replace with
- Function notation: is just like an equation but with replaced by
- Practice makes perfect: The more problems you work through, the more confident you'll become with function operations