Functions
5 minQuiz at the end
What is a Function?
A function is a rule that assigns exactly one output to every input. Think of it as a machine: put in a value (input), get out a value (output).
Notation: f(x) = 2x + 1
- f is the function name
- x is the input (independent variable)
- f(x) is the output (dependent variable)
Evaluating Functions
Substitute the input value:
- f(3) = 2(3) + 1 = 6 + 1 = 7
- f(-2) = 2(-2) + 1 = -3
Domain and Range
Domain: the set of all allowed inputs Range: the set of all possible outputs
For f(x) = xΒ²:
- Domain: all real numbers
- Range: all values β₯ 0 (squares are never negative)
Composite Functions
f(g(x)) means apply g first, then f:
- g(x) = 3x, f(x) = xΒ²
- f(g(2)) = f(6) = 6Β² = 36
Types of Functions
| Type | Example | Graph |
|---|---|---|
| Linear | f(x) = 2x + 1 | Straight line |
| Quadratic | f(x) = xΒ² | Parabola |
| Exponential | f(x) = 2Λ£ | Exponential curve |
| Reciprocal | f(x) = 1/x | Hyperbola |
The Vertical Line Test
A graph represents a function if every vertical line intersects it at most once.