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EDIT#2: Yes, I'm crazy! This IS a function. Thanks for beating the correct logic into me everyone!

I'm using a website provided by my algebra textbook that has questions and answers. It has the following question:

Determine whether the following relation represents a function: $$\{(-9,-3),(2,-1),(7,7),(-1,-1)\}$$

I answered NO, it is not a function but the website says it is. Am I wrong? If so, what am I missing?

EDIT: I was given the following definition in class:

Function: A function is a rule which assigns to each X, called the domain, a unique y, called the range.

My instructor also said that if you plot the points you can tell if it is not a function if it fails the vertical line test. Here is the graph of the above points, and for example it would fail the vertical line test if I drew one on x = 1, right?

enter image description here

Thanks! Jason

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Why do you think it is not a function? –  Chris Eagle Jun 19 '12 at 23:44
Is a domain specified in an earlier part of the question? Some (many?) definitions of function require that the relation be defined on the entirety of the domain in question. –  Austin Mohr Jun 19 '12 at 23:47
@Chris Eagle, I think it is not a function because 1) I have repeating y values (the -1s) and 2) I plotted the points on a graph and it fails the vertical line test. –  Jason Jun 20 '12 at 0:04
The relation (whether or not it is a function) is defined as those particular four points, not as line segments connecting them. Your graph, therefore, should consist of exactly four dots. Does the resulting figure pass the vertical line test? (That said, you are correct that your green graph is NOT a function. That said, if you'd chosen to connect the dots in order of $x$ value, you'd see a function. But, again, the relation described in the problem only involves the dots.) –  Blue Jun 20 '12 at 0:14
@Jason: Uh, you should get rid of the "NOT" from edit #2. –  Zev Chonoles Jun 20 '12 at 5:01
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10 Answers

up vote 11 down vote accepted

All first coordinates are distinct. It's the graph of a function.

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I thought it had to be all second coordinates had to be distinct, not first. Maybe I'm going insane. :) –  Jason Jun 20 '12 at 0:01
Do not connect in the order the points are given. Go from left to right. If the second coordinates were distinct as well, the function would be 1-1. –  ncmathsadist Jun 20 '12 at 0:16
ah I forgot that! thanks for clearing it up! I'm such a newb! –  Jason Jun 20 '12 at 0:22
@Jason: there is a special class of functions where the second coordinates are distinct; these are called “injective”. You might have come across the term elsewhere. (Formally: a function $f$ is injective if $f(x)=f(y)$ implies $x=y$.) –  alexwlchan Jun 20 '12 at 9:30
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It is a function from the set $\{-9,-1,2,7\}$ into a set containing $\{-3,-1,7\}$. As long as each element of the domain, $\{-9,-1,2,7\}$, gets mapped unambiguously to a value (not necessarily distinct), this is a well-defined function.

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You only think it fails the vertical line test at $x=1$ because you drew the graph incorrectly.

You plotted the points you were given, but you also plotted many points that you were not given. You drew a bunch of lines, but there was nothing in the question about lines. The correct graph has four isolated points—the four that were given to you—with nothing in between. Your graph includes points at $(1,1)$ and $(1,-\frac{13}{11})$. But there is nothing in the definition of this function that says it has any values at $x=1$.

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you also plotted many points that you were not given, that's true, I didn't think of it that way. –  Jason Jun 20 '12 at 13:23
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A function doesn't have to be from $\mathbb R$ to $\mathbb R$. The domain of a function can be as simple a set as $\{-9,-1,2,7\}$.

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why not plot the points and see how how the graph looks

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A function cannot have two points that share the same $x$ value.

Your $x$ values are -9, -1, 2 and 7. All your $x$ values are unique (i.e. no repetition), and thus we may conclude that this is indeed a function.

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One way to precisely define a function is as follows: A function is a collection of ordered pairs, no two of which have the same first term. From this definition, it is immediate that your collection of ordered pairs is a function.

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The domain of your function is just 4 discrete numbers. What you do when you draw lines between them is extend the domain to the interval [-9, 7], where every point between the interval's limits would be a member of the set. Like 5, but also 1.25, $\sqrt{2}$ and $\pi$.

BTW, notice that connecting the dots does result in a function, if you first order your set:


and that can't be right; the set {−9, 2, 7, −1} is the same as {−9, -1, 2, 7}.

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As long as every point of domain is related to exactly one point of co-domain(vice versa need not be true),that relation is a function.Two or more points of domain can be related to a single point in co-domain. you can remember it like this: One can have exactly one father, but a father can have one or more children. Here child is a point of domain and father, a point of co-domain. In the given problem, each point of domain has a unique image and hence it is a function.

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It is a function, because every value in the domain has one value in the codomain. You may be confused because you are plotting the points and linking them with lines in the order they are given.

A simple test: are there two $y$ values that have the same $x$ value? If the answer is yes, then the relation is not a function; if the answer is no, then the relation IS a function.

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