Function notation
- f(x) = 3x − 5 means "the function f takes x, multiplies by 3 and subtracts 5". f(4) = 3 × 4 − 5 = 7.
- To solve f(x) = 11 for f(x) = 2x + 3, write 2x + 3 = 11, so x = 4.
- You may also meet composite functions, fg(x) = f(g(x)), and inverse functions, f⁻¹(x), which undo f.
Domain and range
- The domain is the set of input values; the range is the set of output values.
- f(x) = 2x + 1 with domain 0 ≤ x ≤ 4 has range 1 ≤ f(x) ≤ 9.
- f(x) = x² for all real x has range f(x) ≥ 0.
- Some values must be left out of the domain: f(x) = 1/(x − 2) needs x ≠ 2, and f(x) = √(x − 3) needs x ≥ 3.
- For a quadratic, complete the square to find the range: x² − 6x + 11 = (x − 3)² + 2, so the range is f(x) ≥ 2.
Piecewise functions
- A piecewise function uses different rules on different parts of the domain. For example, f(x) = x + 1 for x < 0 and f(x) = 3 for x ≥ 0.
- To sketch it, draw each rule only over its own part of the domain. Check whether the pieces join up.
- To evaluate, choose the rule for that value: here f(5) = 3 and f(−2) = −1.
Key terms
- Function
- A rule that gives exactly one output for each input.
- Domain
- The set of input values of a function.
- Range
- The set of output values of a function.
- Piecewise function
- A function with different rules on different parts of its domain.
- Composite function
- A function made by applying one function then another.
- Inverse function
- A function that undoes another function.