What is a function?
A function is a special relation between two sets. Each input value (from the first set) is joined to exactly one output value (in the second set). You can picture a function as a machine: you feed a number in, the machine follows a rule, and one number comes out.
If the input is x and the rule is "multiply by 3, then add 2", we write the function as f(x) = 3x + 2. Here f(x) is read as "f of x" and stands for the output.
Key idea
In a function, one input can give only one output. Many inputs may share the same output, but one input can never give two different outputs.
Finding the value of a function
To find an output, replace x with the given number and calculate. This is called substitution. Always work out the multiplication or powers before adding or subtracting.
Worked example
Given f(x) = 2x − 5, find f(4).
f(4) = 2(4) − 5 = 8 − 5 = 3.
Now find x when f(x) = 9: 2x − 5 = 9, so 2x = 14 and x = 7.
Recognising a function
Check the arrows in a mapping diagram. If every input arrow points to only one output, it is a function. If any single input points to two outputs, it is not a function. Functions can be one-to-one (each input has its own output) or many-to-one (several inputs share one output).
Remember
- f(x) means the output when the input is x.
- Substitute the number for every x, then simplify.
- One input → exactly one output.