Problem 110 - College Algebra 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 relation in which each input has exactly one output. This can be determined by reviewing the relation, using a vertical line test, or examining the ordered pairs.

Step 1: Understand the Definition of a Function

In mathematics, a function is a rule for taking an input (often a number or a set of numbers) and providing an output (also often a number). The critical characteristic of a function is that for any input, there is exactly one output. So, if you have something that could have more than one response to a single input, that’s not a function.

Step 2: Review the Relation to Determine if it is a Function

Consider the given relation and check each input to see if it maps to one unique output. If you find an input that is attached to more than one output, then this relation is not a function.

Step 3: Using a Vertical Line Test

A relation is a function if and only if no vertical line intersects the graph of the relation at more than one point. This is known as the vertical line test. If any vertical line passes through more than one point on the relation, then it fails the vertical line test and it is not a function.

Step 4: Examining the Ordered Pairs

If a set of coordinate points is given as a relation, then by examining these points we can determine if it is a function or not. If any x-value (input) is repeated with a different y-value (output), this violates the rule for being a function.