February 18, 2026·Pengi AI Team

What Is a Function? Representing with Tables, Graphs, and Mappings

This article explains the mathematical concept of a function, covering its definition as a rule that assigns each input exactly one output. It demonstrates how functions can be represented using tables, graphs, and mapping diagrams, with clear visual examples for each method. Students learning pre-algebra or algebra will find this a clear and accessible introduction to one of the most foundational math concepts.

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Pengi Editor's Note: This article was originally published by Think Academy. We're sharing it here for educational value. Think Academy is a leading K-12 math education provider.

What Is a Function? Representing with Tables, Graphs, and Mappings

By Grade 7 at Think Academy (Grade 8 in school), students learn what functions are and how to represent them with tables, graphs, and mappings. The challenge is not defining a function, but applying the “one input–one output” rule consistently—many confuse domain and range, struggle with table patterns, misread graphs, or get lost in mapping diagrams. This guide explains each representation clearly so those misunderstandings don’t block progress.

What Is a Function?

A function is a special type of relationship between two sets of numbers: an input and an output.

Each input value has exactly one output value.
Different input values can share the same output value.
Inputs are usually called the domain.
Outputs are usually called the range.

What Is a Function? With Input and Output

A function can be thought of as a “machine”: put in an input, the machine follows a relationship, and we always get one output.

Ways to Represent a Function

In the following sections, we will use different methods to represent the same function:

𝑓(𝑥) = 𝑥² − 2𝑥 + 1

Using a Table to Represent a Function

A table lists inputs and outputs side by side.

Using a Table to Represent a Function

Using a Graph to Represent a Function

A graph plots input-output pairs as points on the coordinate plane.

Using a Graph to Represent a Function

The table above would give the points (−1, 4), (1, 0), (2, 1), (3, 4), (4, 9). Connecting them makes the relationship easy to see.

Using Mappings to Represent a Function

A mapping diagram shows arrows connecting inputs to their outputs.

Draw one column of input numbers.
Draw one column of output numbers.
Connect each input to its output with an arrow.

Using Mappings to Represent a Function

This makes it very clear that each input has only one output.

Other Ways to Represent a Function: Ordered Pairs

A function can also be represented as a set of ordered pairs:

{ (−1, 4), (1, 0), (2, 1), (3, 4), (4, 9) }

In each pair, the first number represents the input, and the second number represents the output.

Example Problems: Representing a Function

1.The graph of an equation is shown in the figure. Based on the graph, complete the table of 𝑦 (output) in terms of 𝑥 (input).

Example Problems- Representing a Function

𝑥 (Input)	𝑦 (Output)
1
	4
3
	8

Answer:

𝑥 (Input)	**𝑦 (Output)
1	2
2	4
3	6
4	8

2. Which set of ordered pairs represents 𝑦 (output) as a function of 𝑥 (input)?

A. { (1, 5), (2, 5), (3, 5), (4, 5) }
B. { (5, 1), (5, 2), (5, 3), (5, 4) }
C. { (1, 3), (2, 6), (1, 9), (4, 12) }
D. { (3, 1), (6, 2), (3, 7), (12, 4) }

Answer:

For 𝑦 to be a function of 𝑥, each 𝑥-value must map to exactly one 𝑦-value.

For A: { (1, 5), (2, 5), (3, 5), (4, 5) }, all 𝑥-values (1, 2, 3, 4) are unique, so each 𝑥 maps to exactly one 𝑦.

For B: { (5, 1), (5, 2), (5, 3), (5, 4) }, the 𝑥-value 5 maps to multiple 𝑦-values.

For C: { (1, 3), (2, 6), (1, 9), (4, 12) }, the 𝑥-value 1 maps to multiple 𝑦-values.

For D: { (3, 1), (6, 2), (3, 7), (12, 4) }, the 𝑥-value 3 maps to multiple 𝑦-values.

Only option A satisfies the condition for 𝑦 to be a function of 𝑥.

The answer is A.

Summary: Representing a Function

In a function, different inputs to share the same output (many-to-one) is possible, but a single input can never have two or more outputs.
Functions can be shown in many ways: tables, graphs, mappings, and many other forms.
To test if a relation is a function, check whether each input has one unique output. If on a graph, we can use the Vertical Line Test: if any vertical line crosses the graph at more than one point, then it is not a function.

Not a Function Example

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