We encounter the pairing of quantities very often. For example, the number of gallons of gas and the amount of money needed to purchase it. Or the height of a ball thrown straight up and the time it takes to reach its maximum height.
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
These pairings are best represented as ordered pairs, and a set of ordered pairs is called a relation, where the first elements (x-values) are the domain or independent variables and the second elements (y-values) are the range or dependent variables.
But we are most interested in those relations where no two ordered pairs have the same first element.
Why?
Because a function is a special type of relation in which for each element in the domain there corresponds exactly one element in the range.
All this means is that a function is an equation where every x value is different and doesn’t repeat. This means that a function relates inputs to outputs and each input produces only one output as Math is Fun nicely states.
And we can decide whether a relation is a function in two ways:
- Inspecting each element in the domain and verifying that it only appears once
- Vertical Line Test
The Vertical Line Test is a simple way to determine whether a graph is a function. If we draw vertical lines through a given graph and each line only intersects the graph once then it’s a function. But if each vertical line intersects the graph at more than one point, then the graph is not a function.
Mapping Diagram for a Given Relation
But now that we know that functions are special relations, that means we need to be able to tell them apart.
Introducing… function notation!
A function is given names such as f, g, and h, and they help to identify what is the independent variable versus the dependent variable. Meaning f is function and x is an element in the domain of f.
What is extremely important to note is that these letters (i.e., f, g, and h) can be used interchangeably for y.
For example, we can take our beloved equation y=mx + b and rewrite it as f(x)=mx+b.
Cool!
Together we will find the domain and range of given relations and determine if the relation is a function. We will also use the vertical line test given graphs and tell whether each relation is a function.
Then we will look at the specifics of using function notation and learn how to evaluate all different types of functions.
Relations and Functions – Video
Get access to all the courses and over 450 HD videos with your subscription
Monthly and Yearly Plans Available
Still wondering if CalcWorkshop is right for you?
Take a Tour and find out how a membership can take the struggle out of learning math.