When we write $y=f(x)$ what do we mean? Would we call $y$ a function of $x$ as well? $y=f(x)$ would represent geometrically a curve in the xy-plane, but other than that I don't understand why we would write something like that.
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See Lambda Calculus and functional application. – Mauro ALLEGRANZA Nov 12 '17 at 09:41
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We have the "formal" notation: $f : A \to B$ to denote a function. We have the usual notation $f(x)$ the denote a single-argument function. We have the $f(a)=b$ notation to denote the fact that the $b$ is the value of the function $f$ for argument $a$. And we have traditions: one of them (very old) is that originated with representation of functions with Cartesian coordinates. – Mauro ALLEGRANZA Nov 12 '17 at 09:43
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You can see also the post: What is a function ? – Mauro ALLEGRANZA Nov 12 '17 at 09:44
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See Function: notation and History of the function concept. – Mauro ALLEGRANZA Nov 12 '17 at 09:57
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That depends on part on what $f(x)$ is. Suppose $f(x) = x^2 + 1$. Then, if $y = 5$, we have $f(2) = 5 = y$. But we also have $f(-2) = 5 = y$ as well, so a lot of values of $y$ are matched to two values of $x$. If you're limited to "real" numbers, then some values of $y$ don't match to any values of $x$ at all.
Now suppose $f(x) = x + 1$. This is what you humans like to call a "one-to-one" function. If $y = f(x)$, then each value of $y$ matches to precisely one value of $x$.
That's the only context I've seen that notation in, explaining one-to-one functions, e.g., http://www.mathwords.com/h/horizontal_line_test.htm
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