What is a function?

A function is a special relationship between two sets of numbers (or variables), usually called inputs and outputs, where each input has exactly one output.

• Think of it like a machine:

  1. You put a number in (input)

  2. The function “processes” it

  3. You get one number out (output)

Notation:

• f(x) = output

• x = input

• The equation 2x + 3 tells you how to get the output from the input

Key Terms

How to evaluate a function

Example:

1. If x = 4, then:

✅ Output is 11

2. If x = -1, then:

✅ Output is 1

How to identify a function

A function must assign exactly one output for each input.

• ✅ Examples of functions:

• ❌ Not a function (input gives multiple outputs):

Function representation

1. Equation: f(x) = 2x + 3

2. Table: List inputs and outputs

3. Graph: Plot points (x, f(x)) on a coordinate plane

4. Mapping diagram: Arrows from input → output

Function vs. Relation

• Relation: any set of input-output pairs

• Function: a relation where each input has exactly one output

Vertical line test (for graphs):

• Draw vertical lines on the graph.

• If any vertical line hits the graph more than once, it’s not a function.

Common beginner mistakes

1. ❌ Confusing input and output

2. ❌ Forgetting that each input must have only one output

3. ❌ Misinterpreting tables or graphs

4. ❌ Not using proper function notation

Why functions matter

• They are the language of algebra

• Foundation for linear functions, quadratic functions, and more

• Used in real-world applications: speed, cost, population growth, etc.

Examples of functions in real life

• Temperature conversion: Celsius → Fahrenheit

• Cost of items: f(x) = price × quantity

• Distance over time: d(t) = speed × time