Introduction to Functions

Article Contents

What is a Function?
Function Notation f(x)
Functions as Machines
Ways to Represent a Function
Representation by Formula
Representation by Table
Representation by Graph
Real-Life Examples of Functions
When is a Relation NOT a Function?
Interactive Exercises

1. What is a Function? {#what-is-a-function}

A function is a rule that assigns exactly one output to each input. Think of it as a reliable machine: you put something in, and you always get the same thing out.

Formal Definition

A function $f$ from set $A$ to set $B$ is a rule that assigns to every element $x \in A$ exactly one element $y \in B$ .

We write:

f : A \to B

The key word is exactly one -- for each input, there is one and only one output.

2. Function Notation f(x) {#function-notation}

We use the notation $f (x)$ to describe a function. Here:

$f$ is the name of the function
$x$ is the input (also called the argument)
$f (x)$ is the output (the value of the function at $x$ )

Example

f (x) = 2 x + 3

This means: "take any number $x$ , multiply it by 2, then add 3."

Input $x$	Calculation	Output $f (x)$
$x = 0$	$2 (0) + 3$	$f (0) = 3$
$x = 1$	$2 (1) + 3$	$f (1) = 5$
$x = - 2$	$2 (- 2) + 3$	$f (- 2) = - 1$
$x = 4$	$2 (4) + 3$	$f (4) = 11$

Different Function Names

Functions don't have to be called $f$ . We can use any letter:

g (x) = x^{2}, h (t) = 3 t - 1, p (n) = n + 7

3. Functions as Machines {#functions-as-machines}

A helpful way to think about functions is as a machine:

You feed the machine an input (a number $x$ ).
The machine applies its rule (e.g., multiply by 2, add 3).
The machine produces an output $f (x)$ .

Important: The same input always produces the same output. The machine is consistent.

4. Ways to Represent a Function {#ways-to-represent}

There are three main ways to describe a function:

Representation	Description	Best for
Formula	An equation like $f (x) = x^{2} - 1$	Exact calculations
Table	A list of input-output pairs	Specific values
Graph	A picture in the coordinate plane	Seeing the overall shape

Each representation shows the same function from a different perspective.

5. Representation by Formula {#representation-by-formula}

A formula gives a precise rule for computing $f (x)$ .

Examples

Function	Formula	Type
Linear	$f (x) = 3 x + 2$	Straight line
Quadratic	$f (x) = x^{2}$	Parabola
Direct proportion	$f (x) = 5 x$	Line through origin
Inverse proportion	$f (x) = \frac{6}{x}$	Hyperbola

From a formula, you can compute $f (x)$ for any allowed input.

6. Representation by Table {#representation-by-table}

A table lists selected input-output pairs:

$x$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$f (x) = x^{2} - 1$	$3$	$0$	$- 1$	$0$	$3$	$8$

Tables are useful when:

You only need specific values
You are preparing to draw a graph
The function comes from measured data

7. Representation by Graph {#representation-by-graph}

A graph shows all input-output pairs as points $(x, f (x))$ in the coordinate plane.

The graph gives a visual overview of how the function behaves. Read more in Functions and Graphs.

8. Real-Life Examples of Functions {#real-life-examples}

Functions appear everywhere in daily life:

Temperature during the day

The temperature $T$ depends on the time of day $t$ :

T = f (t)

At 6:00 it might be 10 C, at 14:00 it might be 25 C. Each time has exactly one temperature.

Price of items

If apples cost \ $2 e a c h, t h e t o t a l p r i ce$ P $d e p e n d so n t h e n u mb er o f a ppl es$ n$:

P (n) = 2 n

Distance and speed

A car travelling at 60 km/h covers a distance $d$ that depends on time $t$ :

d (t) = 60 t

Conversion formulas

Converting Celsius to Fahrenheit:

F (C) = \frac{9}{5} C + 32

9. When is a Relation NOT a Function? {#not-a-function}

A relation is not a function if one input produces more than one output.

The Vertical Line Test

On a graph, draw a vertical line at any $x$ -value. If the line crosses the curve more than once, it is not a function.

Example: $x^{2} + y^{2} = 9$ (a circle)

This is not a function because for $x = 0$ , we get $y = 3$ and $y = - 3$ -- two outputs for one input.

10. Interactive Exercises {#interactive-exercises}

Test your understanding of function basics:

Summary

Concept	Description
Function	A rule assigning exactly one output to each input
$f (x)$ notation	$f$ = function name, $x$ = input, $f (x)$ = output
Formula	Algebraic rule, e.g. $f (x) = 2 x + 3$
Table	List of input-output pairs
Graph	Visual picture of all pairs $(x, f (x))$
Vertical line test	If a vertical line crosses the graph more than once, it is not a function

Variables in Functions -- dependent vs independent variables
Functions and Tables -- representing functions with tables
Functions and Graphs -- plotting and reading graphs
Domain of a Function -- which inputs are allowed
Summary of Functions -- quick reference card