Introduction to Functions

Article Contents

1. What is a Function?
2. Function Notation f(x)
3. Functions as Machines
4. Ways to Represent a Function
5. Representation by Formula
6. Representation by Table
7. Representation by Graph
8. Real-Life Examples of Functions
9. When is a Relation NOT a Function?
10. Interactive Exercises

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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:

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

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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

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

Different Function Names

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

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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.

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

There are three main ways to describe a function:

Each representation shows the same function from a different perspective.

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

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

Examples

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

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

A table lists selected input-output pairs:

Tables are useful when:

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

Read more in Functions and Tables.

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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.

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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 $t$:

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 \$2each,thetotalprice$P$dependsonthenumberofapples$n$:

Distance and speed

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

Conversion formulas

Converting Celsius to Fahrenheit:

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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.

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10. Interactive Exercises {#interactive-exercises}

Test your understanding of function basics:

• Variables in Functions
• Reading Function Values
• Function or Not?

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Summary

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Related Articles

• 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