Problem 110 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

A function is a mathematical relationship that assigns exactly one output value to each input value. In the context of ordered pairs, no two pairs in a function have the same first component. A relation is a function if all input values correspond to exactly one output value, or in graphical representation, if it passes the vertical line test.

Step 1: Understanding a Function

A function is a mathematical relationship that assigns exactly one output value to each input value. In terms of ordered pairs, a function can be defined as a set of ordered pairs in which no two different pairs have the same first component.

Step 2: Representing a Relation

In many cases, relations are represented in either tabular form, graphically, or as a set of ordered pairs.

Step 3: Determine if the Relation is a Function

Now examine the relation. If any input value (or 'x-value') corresponds to two or more output values (or 'y-values'), then the relation is not a function. If every input value corresponds to exactly one output value, then the relation is a function.

Step 4: Using the Vertical Line Test for Graphs

For relations presented as graphs, the vertical line test can be employed. This test states that if any vertical line drawn through the graph intersects the graph at more than one point, then the graph is not a function. If every vertical line intersects the graph at exactly one point, then the graph represents a function.