🔤 What Is a Function?
A function is a relationship where each input has exactly one output.
In math terms: a function f maps every element in the domain to exactly one value in the range.
Example: f(x) = x + 2
If x = 3, then f(3) = 3 + 2 = 5
🧠 Real-Life Analogy
Think of a function like a vending machine:
- You press a specific button (input)
- You get one snack (output)
- Each button gives only one snack — that's what makes it a function
📌 Domain and Range
Domain: The set of all possible inputs (x-values)
Range: The set of all possible outputs (f(x) or y-values)
Example: f(x) = √x → Domain: x ≥ 0, Range: y ≥ 0
📝 Function Notation
- Instead of using y = ..., we write f(x) = ...
- Example: f(x) = 2x + 3
- g(x) = x² is read as "g of x equals x squared"
✅ How to Evaluate Functions
Just plug in a value for x.
Example: f(x) = 3x - 1. Find f(4):
f(4) = 3(4) - 1 = 12 - 1 = 11
🧩 Types of Functions
- Linear: f(x) = mx + b (straight line)
- Quadratic: f(x) = ax² + bx + c (parabola)
- Cubic: f(x) = ax³ + bx² + cx + d
- Exponential: f(x) = a·b^x
- Absolute Value: f(x) = |x|
- Square Root: f(x) = √x
- Rational: f(x) = 1/x or similar
- Step / Greatest Integer: f(x) = ⌊x⌋
📉 Graphing Functions
Each function has a unique graph:
- Linear → straight line
- Quadratic → parabola (U-shape)
- Exponential → curves that grow rapidly
Use a table of values or graphing calculator to sketch graphs.
🔁 Inverse Functions
An inverse function undoes the original function.
If f(x) = 2x + 1, then f⁻¹(x) = (x - 1)/2
Only functions that are one-to-one (pass the horizontal line test) have inverses that are also functions.
🧬 Composite Functions
These are functions inside functions. Written as (f ∘ g)(x) = f(g(x))
Example: f(x) = x², g(x) = x + 3 → f(g(x)) = (x + 3)²
🔢 Piecewise Functions
These are functions defined by different expressions depending on the input value.
f(x) = {
x + 2 if x < 0
x² if x ≥ 0
}
Useful for modeling situations with different rules in different intervals.
🔎 Vertical Line Test
A graph represents a function if a vertical line crosses it at most once at any point.
🧠 Common Function Questions
- Is this relation a function?
- What’s the domain and range?
- Evaluate f(2)
- What kind of function is this?
- Graph f(x) = 1/x
- Find the inverse of f(x)
- Compose two functions
📝 Practice Problems
- f(x) = 2x + 5. Find f(3)
- What is the domain of f(x) = 1/(x - 2)?
- Is the graph of f(x) = x² a function? Why?
- If f(x) = x² and g(x) = x + 1, find f(g(2))
- Graph f(x) = |x - 3|