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If there is a variable "$x$" , and another is "$y$", and let's say I know the relation between them as

$$x= y+1$$

Now, I want to use them in the explanation of some mathematical theory/proof/equation etc. Why a different notation called $f(x)$ ( function of $x$ ) is used to represent $y$ ? Why can't y itself be used ? Cannot i just remember that "$y$" has some relation with "$x$" . Is their some other importance also, other than representing a relation of "$y$" with "$x$", when some term like $f(x)$ is used ?

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Early on, the notion of 'functions' are introduced in much more detail and rigor than the notion of 'dependent variables'. Thus, there is a tendency to phrase things in terms of functions rather than in terms of dependent variables. –  Hurkyl Jan 25 '13 at 19:31

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They're essentially the same thing, except for a couple of points. Writing y = x - 1 obfuscates the relationship between y and x. Does y depend on x or is y a constant? Is x a constant or is it a variable? It's not clear just by writing y = x - 1 what exactly you mean. On the other hand, inserting the variable makes it transparent what's going on. Now you might just be tempted to say "assume y is always considered a function of x and suppress variable notation". Fine. What about y = xz - 2z + x - 1? Does y depend on x and z? Does z depend on any other parameter? It quickly becomes confusing what you mean.

I suppose nothing is stopping you from writing y(x) = x - 1 and inserting the dependency. It's done in some contexts. But then you're just using a different letter for f, which doesn't seem much different. Also, f is a stand in for "function", I'd guess.

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In y = xz - 2z + x - 1 , we can say y = f(x,z) ?? ie. value of y depends upon both, x as well as z ? –  Vishwas Gagrani Jan 25 '13 at 19:29
    
If y depends on x and z, sure. But maybe y doesn't depend on either. Or only one. Or maybe z depends on x too so we have f(x, z(x)). The point is having this notation helps clarify what exactly is intended. –  anonymous Jan 26 '13 at 0:19

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