Function notation
- A function takes an input and gives an output. f(x) = 3x + 2 means 'multiply by 3, then add 2'.
- f(4) means substitute x = 4: f(4) = 3 × 4 + 2 = 14.
- To solve f(x) = 20, write 3x + 2 = 20, so x = 6.
Composite functions
- fg(x) means do g first, then f. If f(x) = 3x + 2 and g(x) = x², then fg(x) = 3x² + 2 and gf(x) = (3x + 2)².
- fg(2) = f(g(2)) = f(4) = 14. Usually fg(x) and gf(x) are different.
Inverse functions
- The inverse function f⁻¹(x) reverses f. Write y = f(x), make x the subject, then swap back to x.
- For f(x) = 3x + 2: y = 3x + 2, so x = (y − 2)/3, and f⁻¹(x) = (x − 2)/3.
- The inverse undoes the function: f(4) = 14 and f⁻¹(14) = 4.
Key terms
- Function
- A rule that turns each input into exactly one output.
- Input
- The value that goes into a function.
- Output
- The value that comes out of a function.
- Composite function
- Two functions applied one after the other, such as fg(x).
- Inverse function
- The function that reverses another, written f⁻¹(x).