Does $y = f(x)$ and $y=g(x)$ imply $f(x)=g(x)$? Okay, this maybe a very dumb question but I can't seem to find a "proper" reason to show why this isn't true.
In some calculus books or notes, I have come across questions where they sometimes begin by saying, "If $y =f(x)$ and $y =g(x)$ are two functions and blah-blah..." I'm curious, is this some "abuse of notation" thing because it should be clear from the context? Because should it not mean that $y =f(x) = g(x)$?.
There is also another context where this is used— While making graphs. The question usually says graph $y=f(x)$ and $y=g(x)$ etc.
Surely it is not a substitution, otherwise the two functions would be same. So what does equating (possibly multiple) functions to $y$ mean? Or what is the intended meaning when we say $y =f(x)$ and $y =g(x)$?
 A: tl; dr: A function is a deterministic relation between two quantities. Strictly, we should say, "If $f$ and $g$ are two functions and blah-blah...," and direct students to "graph $f$ and $g$."

In the modern formulation, if $X$ and $Y$ are sets, a function $f:X \to Y$ is a subset $f \subset X \times Y$ of the Cartesian product satisfying the condition,

For every $x$ in $X$, there exists a unique $y$ in $Y$ such that $(x, y) \in f$.

If $(x, y) \in f$, we write $y = f(x)$ and say $y$ is the value of $f$ at $x$. Loosely, each "input" $x$ to the function $f$ yields a unique "output" $y = f(x)$. The letters $x$ and $y$ are, as noted in your (William's) answer, local dummy variables used to denote elements of the domain $X$ and the codomain $Y$, respectively.
Particularly, if symbols signify real numbers for definiteness, then
$$
y = x^{2},\qquad
u = s^{2},\qquad
\clubsuit = \heartsuit^{2},\qquad
\text{(etc., etc.)}
$$
have identical meaning as notations for the real squaring function $\square \mapsto \square^{2}$. (The square may be viewed as a symbol, but better, it may be viewed as a blank in a web form where we can input any numerical expression, and the second square autofills.)
To the edited question, "Let $y = f(x)$ and $y = g(x)$" has meanings that include "Let $f$ and $g$ be functions" and "Let $x$ and $y$ be real numbers satisfying $y = f(x)$ and $y = g(x)$." In the second interpretation, we do have $f(x) = g(x)$.

To get tangentially (as it were) nitpicky, we should also not write $(x^{2})' = 2x$ and the like: $x^{2}$ is not a function but an expression representing a numerical quantity (assuming $x$ is a number), and so does not have a derivative. Instead, we should say,

"If $f(x) = x^{2}$ for all real $x$, then $f'(x) = 2x$ for all $x$."

Because that's cumbersome, we often fall back on Leibniz notation, $\frac{d}{dx}(x^{2}) = 2x$, with the understanding that we Really Mean the implication in quotes.
A: I asked the question and this is my understanding of what is going on here.
I have always understood the notation $f(x)$ to mean the image of $x$ under $f$ where $x$ is a stand-in for all the elements in the domain of $f$.
Which also, means the $x$ in $f(x)$ and the $x$ in $g(x)$ are different $x$'s because they are a stand-in for different sets of elements. The former $x$ represents any element in the domain of $f$ while the latter $x$ is any element in the codomain of $g$.
Similarly, the $y$ in $y =f(x)$ is stand-in for set of all the elements in the range (not codomain, mind you). So, the $y$ in $y = f(x)$ and the $y$ in $y =g(x)$ are different because they represent different sets of elements. The former $y$ represents any element in the range of $f$ while the latter $y$ is any element in the range of $g$.
That's how I understand it, at least. Thinking of $x$'s and $y$'s as not one particular elements but rather a "stand-in" or a dummy variable that is suppose to represent all the elements in the domain or range respectively, seems like what that notation appears to suggest. I'm curious to see what everyone else thinks of it.
