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How can two different domains map to the same range in a function? I'm honestly trying to become better at math, so sorry if I come off as ignorant. A function must have every domain value associate to only one y-value. Ok, that makes sense. But then, in a function, one y-value IS able to map to 2 different domains. Why? This puzzles my logic. A function has a rule, so if, per say, my rule is f(x)=x+1 , then I can have the following set: {(1,2),(2,3)}. This set follows the function, however, I am unable to have a set such as: {(1,2),(1,3)}, ok so that makes sense according to my rule. However, a function CAN have a relation of the following: {(1,2),(2,2)}, this is allowed in a function, however, I don't see how this could ever follow my rule? Is there an example where my brain could be proved otherwise? lol Thanks guys!

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  • $\begingroup$ The most basic example that comes to mind is a constant function such as $f(x)=5 , x \in \mathbb{R}$. $\endgroup$ – 2chromatic Jan 27 '18 at 1:04
  • $\begingroup$ $f~:~\{-1,1\}\to\{1\}$ given by $f(x)=x^2$ satisfies $(-1)^2=1$ and $1^2=1$ is an example of where $f(-1)=f(1)$. Perhaps you are only trying linear functions of the form $ax+b$ with $a\neq 0$. As for $\{(1,2),(2,2)\}$, we might call such a function a "constant function" and might be written simply as $f(x)=2$, or if you insist on a more convoluted way, how about $f(x)=|x-1.5|+1.5$ where one has $f(1)=f(2)=2$. $\endgroup$ – JMoravitz Jan 27 '18 at 1:04
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    $\begingroup$ Here is another example of a function which is not one-to-one: Denote the function "$\text{First}$" with domain being the set of words in the English language and range the set of letters used in the English alphabet defined as taking a word as input and outputting the first letter of that word. We have $\text{First}(apple)=a=\text{First}(ant)$. Written in set form we have $\text{First}=\{(aardvark,a),(aarnet,a),\dots,(first,f),\dots,(zulu,z)\}$. We don't even need our functions to have "rules" that "make sense" in the first place, $\{(1,\pi),(potato,5)\}$ can be a perfectly valid function $\endgroup$ – JMoravitz Jan 27 '18 at 1:19
  • $\begingroup$ Nice way to prove the point @JMoravitz $\endgroup$ – chromechris Jan 27 '18 at 1:31
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  • I think intuitively the idea is A function takes its input and follows a rule to produce its output. Sometimes, following the rule for different inputs leads to the same output result.

  • You may be used to functions where different inputs always produce different outputs. For example, "adding one to a number" is a function where different inputs produce different outputs. Or "doubling a number".

    However, some rules can behave differently: even if you start with two different inputs, the rule may transform them into the same output.

  • Here's an example: a function like $f(x)=x^2$ sends every number to its square. So, for example, $f(3)=3\cdot 3 = 9$. Because (-3)(-3) = +9, we know that $f(-3)=9$ as well. So, squaring -3 and squaring 3 end up as the same result.

    So, the function is built of the tuples $(3,9)$ and also $(-3, 9)$.

  • Or you can consider the rule $g(x)$ which takes in a number and tells you whether it's odd or even. Then $g(2)=g(4)=g(6)=\ldots = \mathsf{even}$ and $g(1)=g(3)=g(5)=\ldots = \mathsf{odd}$. Many inputs lead to the same result.

  • Functions can even deal with things other than numbers. Imagine a function whose inputs are people and whose outputs are numbers— the person's age. Then this rule may associate two different people with the same age.
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Functions are designed to model specific relationships, where we associate with each input ("element of the domain") exactly one output ("element of the range"). There is a certain amount of asymmetry in this definition; nothing prohibits a function from sending several inputs to the same output (but fear not, we have such a concept in mathematics as well! Functions that do send different inputs to different outputs are called "one-to-one" or "injective").

It is not an accident that (given a function $f$, and an element $x$ in the domain of $f$) we generally say "$f$ of $x$" when we see $f(x)$. Lots of relationships in the real world are functions in the mathematical sense, and this is especially clear if we can say "the $\_\_\_$ of $\_\_\_$".

For example, "the (biological) mother of $x$" is a function from the set of all people to the set of all people, that associates with a person $x$ that person's mother. Any person has exactly one biological mother, and so it satisfies the definition of a function (each input person gets paired with exactly one output person).

If we denote this function by $f$, then for a given woman $y$, the equation $f(x) = y$ may have many different solutions; a woman may have more than one child, and for any child $x$ of mother $y$, "the mother of $x$ is $y$" is true (symbolically, $f(x) = y$). This is an example where different inputs (people) may get sent to the same output (their mother).

Here again we see asymmetry: If we had tried to define our function by "the child of $x$", that sends to each biological mother her child, this would not be a function: A given input (biological mother) may correspond to several outputs (that mother's children). It certainly constitutes a relationship between biological mothers and people, but in general, we haven't specified any criteria that makes the outputs unique.

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  • $\begingroup$ Nice! Hmm, thanks to all who have chimed in! I was wondering, my notion of equations has changed, lol. At first I saw functions ONLY as equations which take input and give an output, which is not wrong, but I did not understand the conceptual constraints on such. Now that I do, I'm left wondering at what the significance of splitting off equations such as f(x)=y^2+3 (non-function relation), and f(x)=x(0)+3 (function relation)? What is the purpose/importance/practice of a function vs a non-function. Why prefer a function over just a relation and vice-versa? $\endgroup$ – chromechris Jan 27 '18 at 1:58
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    $\begingroup$ Both functions (single output for every input) and relations are useful, and I don't really have a great answer to your question. But, functions eliminate choice. We like deterministic things, and functions are deterministic: You give a function the same input, and it'll always give you the same output. But for a relation, we might have to choose between several "outputs" for a given input. The first way is more "streamlined" in that we don't have to make choices, we just follow a single path. But the second way allows us to model more things. $\endgroup$ – pjs36 Jan 27 '18 at 2:37
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    $\begingroup$ Awesome explanation! Thanks @pjs36 $\endgroup$ – chromechris Jan 27 '18 at 2:40

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