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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 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 by step solution

01

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鈥檚 not a function.
02

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.
03

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.
04

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.

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Most popular questions from this chapter

What does it mean if a function \(f\) is increasing on an interval?

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ x^{2}+(y-2)^{2}=4 $$

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} (x-3)^{2}+(y+1)^{2} &=9 \\ y &=x-1 \end{aligned} $$

If \(f(x+y)=f(x)+f(y)\) and \(f(1)=3,\) find \(f(2), f(3)\) and \(f(4) .\) Is \(f(x+y)=f(x)+f(y)\) for all functions?

Exercises \(98-100\) will help you prepare for the material covered in the first section of the next chapter. In Exercises \(98-99,\) solve each quadratic equation by the method of your choice. $$ -x^{2}-2 x+1=0 $$
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