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I do not understand why functions are needed in mathematics(algebra, to be exact). They are nearly the same as regular equations. Yeah, "there should be only one output for a particular input", and so what? Where is this useful? Textbooks do not tell anything about it. Explain me, please. Thank you, that will be really helpful for anyone who has the same problem as me. I tried to search in internet for the answer, but I didn't find it.

Tim Solnze
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    Functions are certainly not "nearly the same as regular equations". – Qi Zhu Aug 11 '19 at 06:20
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    You sound like somebody who wonders what's the point of a wife... then concludes they're nearly the same as regular.. ;) – AgentS Aug 11 '19 at 06:32
  • "One output" could as well be a vector – dEmigOd Aug 11 '19 at 06:32
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    We always want to predict how the value of some quantity (such as the position of an object) depends on some other quantity (such as time). But that means we are already talking about functions. – littleO Aug 11 '19 at 06:49
  • Functions are e.g. very useful if we want to mod out irrelevant information. An example that you will meet daily in real life is the function $wd:\text{Days}\to{\text{ma, tu, we, th, fr, sa, su}}$. – drhab Aug 11 '19 at 06:59
  • A function isn’t an equation. An equation is $f(x)=x^2$; a function is just the $f:\mathbb R\to\mathbb R$ part. – gen-ℤ ready to perish Aug 11 '19 at 08:33

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Consider the ambient air temperature at midday, in a spot in the shade outside your house. It has a particular value (only one). The temperature in that same spot will change as the day goes on, but it will only ever have a single value at any given time. For example, it might be 20 degrees at noon and 10 degrees at 6 pm. It cannot be both 10 degrees and 20 degrees at 6 pm.

You could say that the ambient air temperature in a spot in the shade outside your house is a function of time. That's why it's useful to talk about functions, because they can be used to describe how something (e.g. a measurement) changes with respect to something else (e.g. time). Loosely speaking, they give an unambiguous value for a given input.

Now consider a different type of "machine" (or mathematical operation) which can give multiple outputs for each input. It might also be useful to us. But we can also think of these as functions, because we can instead define a function whose output is always one tuple of values. For example:

$f(x) = (x,x+1,x+2)$

is a function which returns one value. The value is the tuple $(x,x+1,x+2)$. The tuple contains 3 values, but that's fine, because they are wrapped up inside of one thing (namely the tuple). So you see that the definition of a function is quite flexible.

In summary, functions are a useful type of operation to define because:

i) They make it easy to unambiguously describe things which change with respect to something else

ii) Their definition is flexible enough to accommodate more complicated things

If we learn about the general properties of functions, due to (i), we will learn a lot about how to mathematically describe common phenomena. Due to (ii), we will be able to treat our knowledge of functions like a warm blanket, often retreating back into that familiar way of thinking when things get uncomfortably complicated.

GingerBreadMan
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Functions don't have any intrinsic point, other than as a convenient shorthand for saying "a mathematical object that produces an output, when given an input". If you want to rigorously talk about equations and you never find yourself saying things like the above, then there is no need to work with functions.

The above paragraph is a little tongue-in-cheek, because of course the most interesting equations are equations between "mathematical objects that produce an output, when given an input". You just might not need to refer to these objects by such a name, if they already have other names: like $x^2$, $1/(1+x)$, $\sin x$, etc.

pre-kidney
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