There are different ways of looking at functions. We will consider a few. But first, we need to discuss some terminology.

A "relation" is just a relationship between sets of information. Think of all the people in one of your classes, and think of their heights. The pairing of names and heights is a relation.

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In relations and functions, the pairs of names and heights are "ordered", which means one comes first and the other comes second. To put it another way, we could set up this pairing so that either you give me a name, and then I give you that person's height, or else you give me a height, and I give you the names of all the people who are that tall.

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The set of all the starting values in a relation is called "the domain" and the set of all the ending values is called "the range." The domain is what you start with; the range is what you end up with; the domain is the *x*'s, the range is the *y*'s. (I'll explain more on the subject of domains and ranges later.)

A function is a "well-behaved" relation, by which we mean that, given a starting point, we know the exactly one ending spot to go to; given an *x*-value, we get only and exactly one corresponding *y*-value. (Note: This means that, while all functions are relations [since functions do pair information], *not* all relations are functions. Well-behaved family members are a subset of all your relations; so also functions [being well-behaved] are a subset of all math relations.)

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Let's return to our relation of your classmates and their heights, and let's suppose that the domain is the set of everybody's heights. Let's suppose that there's a pizza-delivery guy waiting in the hallway; all the delivery guy knows is that the pizza is for the student in your classroom who is five-foot-five.

Now let the guy in. Who does he go to? What if nobody is five-foot-five? What if there are *six* people in the room that are five-five? Do they all have to pay? What if you are five-foot-five? And what if you're out of cash? And allergic to anchovies? Are you still on the hook? Ack! What a mess!

The relation "height indicates name" is not well-behaved. It is not a function. Given the relationship (*x*, *y*) = (five-foot-five person, name), there might be six different possibilities for *y* = "name". For a relation to be a function, there must be *only and exactly* one *y* that corresponds to a given *x*.

The above shows a function. You can tell that this is a function by tracing from each *x* to each *y*. There is only one *y* for each *x*; there is only one arrow coming from each *x*.

Ha! Bet I fooled some of you on this one! This *is* a function! There is only one arrow coming from each *x*; there is only one *y* for each *x*. It just so happens that it's always the same *y* for each *x*, but it is only that one *y*. So this is a function; it's just an extremely *boring* function!

This one is not a function: there are *two* arrows coming from the number 1; the number 1 is associated with two *different* range elements. So this is a relation, but it is not a function.

Okay, this one's a trick question. Each element of the domain that has a pair in the range is nicely well-behaved. But what about that 16? It *is* in the domain, but it has no range element that corresponds to it! This won't work! So then this is not a function. Heck, it ain't even a relation!

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The Vertical Line Test provides a graphical way of testing whether a relation is a function. If there is no place on the graph where a vertical (that is, up-and-down) line would cross twice, then the relation passes the test, and is a function. Now, what does all that mean?

Looking at this function stuff graphically, what if we had the relation that consists of a set containing just two points: {(2, 3), (2, −2)}? We already know that this is not a function, since *x* = 2 goes to each of *y* = 3 and *y* = −2.

If we graph this relation, it looks like:

Notice that you can draw a vertical line through the two points, like this:

This characteristic of non-functions was noticed by I-don't-know-who, and was codified in "The Vertical Line Test": Given the graph of a relation, if you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function.

Consider the graph below:

This graph shows a function, because there is no vertical line that will cross this graph twice.

Now consider the ellipse graphed below:

This graph does not show a function, because any number of vertical lines will intersect this oval twice. For instance, the *y*-axis intersects (crosses) the line twice.

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The quickest way to determine if a given formula or equation represents a function is to look at your calculator. If the formula can be entered into the calculator (so the calculator can graph it or evaluate it or whatever), then it's a function; otherwise, not. What does this mean?

Thinking of all the graphing you've done, the simplest method for a given equation is to solve for "*y*=", make a T-chart, pick some values for *x*, solve for the corresponding values of *y*, plot your points, and connect the dots, yadda, yadda, yadda.

Not only is this useful for graphing, but this methodology gives yet another way of identifying functions: If you can solve for "*y*=", then it's a function. In other words, if you can enter it into your graphing calculator, then it's a function. The calculator can only handle functions. For example, 2*y* + 3*x* = 6 is a function, because you can solve for *y*:

2*y* + 3*x* = 6

2*y* = −3*x* + 6
*y* = (−3/2)*x* + 3

On the other hand, *y*^{2} + 3*x* = 6 is not a function, because you can not solve for a *unique* *y*:

I mean, yes, this is solved for "*y* =", but it's not *unique*. Do you take the positive square root, or the negative? Besides, where's the "±" key on your graphing calculator? So, in this case, the relation is not a function.

By the way, you can also check this relation by using our first definition from above. Think of plugging in "*x* = −1". Then we get *y*^{2} − 3 = 6, so *y*^{2} = 9, and then *y* can be either −3 or +3. That is, if we did an arrow chart, there would be two arrows coming from *x* = −1. This again shows that the relation is not a function.

URL: https://www.purplemath.com/modules/fcns.htm

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