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Chapter 2: Problem 110

Explain how to determine whether a relation is a function. What is a function?

Short Answer

Expert verified
A function is a mathematical concept where each input value is related to exactly one output value. A relation can be determined to be a function if each input is paired to exactly one output. This can be determined through several methods, such as by evaluating lists of points, using the vertical line test on a graph, or by assessing a mapping diagram. If you find any input value that is paired with more than one output value, the relation is not a function.

Step by step solution

01

Definition of a function

Firstly, it is important to understand what a function is. In mathematics, a function is a rule that assigns to each input exactly one output. The set of all possible inputs is called the domain, and the set of all possible outputs is called the range.
02

Identifying a function in a list or set of ordered pairs

A function can be represented in several different ways, including graphs, equations, sets of ordered pairs, and mapping diagrams. If the relation is given as a set of ordered pairs, each first element (usually denoted as \(x\)) represents an input of the function and the second element (usually denoted as \(y\)) is the output related to that input. In order to determine if the relation is a function using this method, check whether each input \(x\) is paired with exactly one output \(y\). If any input is paired with more than one output, then the relation is not a function.
03

Identifying a function using a graph

If the relation is represented as a graph, one could use the vertical line test to check if the relation is a function. This test states that a relation is a function if and only if every vertical line crosses the graph at most once. That means that for every value of \(x\), there is exactly one value for \(y\).
04

Identifying a function using a mapping diagram

With a mapping diagram, the relation is displayed as two sets of numbers or objects. Arrows link the elements from one set to another. To check whether the relation is a function or not, ensure there are no inputs that maps to more than one output.

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