Eighth Grade Mathematics: Definition of a Function: Domain, Range, and Mapping
The concept of a function is one of the unifying ideas of modern mathematics, serving as the mathematical description of cause and effect. Whether tracking how the temperature of an oven changes over time, how the cost of postage depends on the weight of a package, or how the speed of a dropped object increases with every second of free fall, functions formalize predictable relationships. In this chapter, you will master the formal mathematical definition of a function, investigate inputs and outputs, analyze domains and ranges, and evaluate mapping diagrams.
The Unambiguous Rule of a Function
In mathematics, a relation is simply any set of ordered pairs connecting inputs to outputs. However, a function is a very special, strictly disciplined kind of relation:
Formal Definition of a Function:
A function is a relation that assigns to each input value in the domain
EXACTLY ONE output value in the range.
The critical phrase in this definition is exactly one. An input can never produce two different outputs. If you press a single button on a soda vending machine, you expect exactly one specific beverage to drop into the slot; if pressing the cola button sometimes delivered a cola and other times delivered an orange soda, the machine would be malfunctioning and unpredictable. In the exact same way, a mathematical rule that assigns multiple outputs to a single input is not a function.
The Function Machine Analogy:
Input (x)
|
v
+-----------------+
| Function f(x) | <--- Predictable mathematical rule
+-----------------+
|
v
Output (y)
Notice that the reverse condition is NOT required: different inputs ARE permitted to produce the exact same output. For example, squaring the input 3 produces 9, and squaring the input -3 also produces 9. That is completely valid for a function, because each individual input still leads to only one output.
Domain and Range
Every function operates between two well-defined sets of numbers:
Domain: The complete set of all allowable INPUT values (x-coordinates).
Range: The complete set of all resulting OUTPUT values (y-coordinates).
When given a finite set of ordered pairs, such as {(1, 4), (2, 7), (3, 10), (4, 13)}, the domain is simply the set of first elements {1, 2, 3, 4}, and the range is the set of second elements {4, 7, 10, 13}.
Continuous versus Discrete Domains
Domains are classified into two broad categories based on the nature of the independent variable:
A discrete domain consists of isolated, individual values with gaps between them, typically counted in whole numbers (such as the number of tickets purchased or the count of passengers on a bus). Discrete relations are graphed as individual, disconnected dots.
A continuous domain consists of all real numbers within a specified interval without any gaps, typically representing measured physical quantities like time, distance, mass, or temperature. Continuous relations are graphed as solid, unbroken lines or curves.
Visualizing Functions with Mapping Diagrams
A mapping diagram illustrates a relation by drawing two parallel columns or ovals representing the domain and range, with directed arrows connecting each input to its corresponding output.
Function (Valid): NOT a Function (Invalid):
Domain (x) Range (y) Domain (x) Range (y)
+-------+ +-------+ +-------+ +-------+
| 1 |------->| 5 | | 2 |------->| 4 |
| 2 |------->| 8 | | |------->| 9 | <--- 2 has TWO arrows!
| 3 |------->| 11 | | 5 |------->| 15 |
+-------+ +-------+ +-------+ +-------+
Each input has EXACTLY ONE arrow. Input 2 has MULTIPLE outputs!
To determine whether a mapping diagram represents a function, inspect the domain column. If every input element has exactly one arrow pointing away from it, the relation is a function. If any input has two or more arrows pointing away from it, the relation is not a function.
Chapter Exercises
Exercise 1: Determine whether the relation {(2, 5), (3, 8), (4, 11), (2, 9), (5, 14)} is a function. State the domain and range of the relation and explain your reasoning in paragraph form.
Exercise 2: A relation is defined by the rule y = x^2 - 3 with the restricted domain {-2, -1, 0, 1, 2}. Calculate the corresponding range values, write the relation as a set of ordered pairs, and determine whether it represents a function.
Exercise 3: Explain in paragraph form why a table tracking the heights of all students in an eighth-grade classroom represents a function of student name to height, but would not necessarily represent a function of height to student name.
Solutions and Explanations
Solution 1: To determine whether the relation {(2, 5), (3, 8), (4, 11), (2, 9), (5, 14)} is a function, inspect the input values (the x-coordinates). The input 2 appears twice: once paired with the output 5, and once paired with the output 9. By definition, a function cannot assign more than one output to any single input. Because the single input 2 corresponds to two different outputs (5 and 9), this relation fails the definition of a function. The domain is the set of all input values, which is {2, 3, 4, 5}. The range is the set of all output values, which is {5, 8, 9, 11, 14}.
Solution 2: To find the range for the function y = x^2 - 3 with domain {-2, -1, 0, 1, 2}, substitute each domain element into the equation. For x = -2, y = (-2)^2 - 3 = 4 - 3 = 1. For x = -1, y = (-1)^2 - 3 = 1 - 3 = -2. For x = 0, y = (0)^2 - 3 = 0 - 3 = -3. For x = 1, y = (1)^2 - 3 = 1 - 3 = -2. For x = 2, y = (2)^2 - 3 = 4 - 3 = 1. The set of ordered pairs is {(-2, 1), (-1, -2), (0, -3), (1, -2), (2, 1)}. The unique outputs forming the range are {-3, -2, 1}. This relation is a valid function because every single input value in the domain is paired with exactly one output value. The fact that the inputs -2 and 2 both produce 1, and -1 and 1 both produce -2, is completely permissible in a function.
Solution 3: In an eighth-grade classroom where the input is a student's name and the output is that student's physical height, the relationship is a function because every individual student has exactly one unique height at any given moment. An input of Sarah can produce only one measured height, such as 62 inches; it is impossible for Sarah to simultaneously be 62 inches tall and 58 inches tall. However, if you reverse the relationship so that height is the input and student name is the output, the relation is no longer guaranteed to be a function. In a typical classroom, multiple students often share the exact same height; for example, if both Sarah and Marcus are 62 inches tall, the single input of 62 inches would point to two distinct outputs (Sarah and Marcus). Because an input would correspond to multiple outputs, the reverse relation is not a function.