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.

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 Graph to Represent a Function

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

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.

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).

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.

Want more printable practice?

Additional Math Topics for Grade 8 – with Free Worksheets

• How to Solve Equations with Variables on Both Sides in 3 Simple Steps
• How to Solve Equations with Parentheses in 4 Simple Steps
• How to Solve Equations with Fractions and Decimals in 3 Simple Steps
• Identifying Linear vs. Nonlinear Functions
  

About Think Academy

Think Academy, a leading K–12 math education provider wholly owned by TAL Education Group, is dedicated to helping students build strong mathematical foundations and critical thinking. Our structured curriculum provides multiple course levels designed to accommodate students with diverse academic goals and proficiency levels, ensuring targeted and effective learning experiences. Supported by advanced teaching methods, expert instructors, and innovative AI technology, Think Academy consistently demonstrates excellence, trustworthiness, and proven expertise in mathematics education.

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