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I understand the phrase $y=f(x)$ to mean $y$ is assigned the value of $f$ evaluated at each point $x$, when we have two functions we might write $y=f(x)$ and $y=g(x)$ for some functions $f, g$. Now we can evaluate the functions at some point, lets say $x=1$ to give $y=f(1)$ and $y=g(1)$ however its not neccesary for $f(1) = g(1)$ (think of any two functions who do not agree at $x=1$), so we cannot write $y = f(1) = g(1)$ as $y$ appears to take two different values (namely $f(1)$ and $g(1)$). Could someone try explain my misunderstanding to me please? I am high-school aged. EDIT I have made sense of the problem like this, we have been using $x$ as a variable for so long and never question why $x$ takes different values because we know its value isnt fixed, so the same is true for $y$....)

Nav Bhatthal
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  • I think its dodgy because, if $y=g(x) = f(x)$ this only holds if $x$ is a point of intersection of the graphs, or if $f=g$ for all $x$, In general after substituting $x$ for an element of the domain of both functions we get two values for $y$ but we know $y$ cannot be two things at once..... – Nav Bhatthal Jul 31 '23 at 12:55
  • The question posted as " Misleading function notation" only furthers my confusion, – Nav Bhatthal Jul 31 '23 at 12:56
  • If $f(x)$ and $g(x)$ are different functions, DON'T write $y=f(x)$ and $y=g(x)$ inside the same discussion ... for example, use different letters $y=f(x)$ and $z=g(x)$. – Ned Jul 31 '23 at 13:10
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    I have seen plenty of times a question which asks, "Plot $y=x^2$ and $y=x$ on the same coordinate axis" for example. – Nav Bhatthal Jul 31 '23 at 13:13
  • Yes, the answers in that linked question don't say what is important. The thing is that when one writes "... the functions/graphs $y=f(x)$ and $y=g(x)$", it should be understood as $f={(x,y)\in\mathbb{R}^2\mid y=f(x)}$ and $g={(x,y)\in\mathbb{R}^2\mid y=g(x)}$. The variables are bound. Their scope is withing the definition of the graph/function being talked about. The $x$ and $y$ in the first are in a different scope as those in the second. They are different variables. – NDB Jul 31 '23 at 13:22
  • This is what my answer in the linked question actually says @NDB – nicomezi Jul 31 '23 at 13:24
  • I see, like when I write $y= g(1)$ I am saying that the graph of $g$ passes through $(1,g(1))$, $g(1) = y$ has nothing to do with the $y$ in the definition of $f(x)$, you see? – Nav Bhatthal Jul 31 '23 at 13:24
  • @nicomezi No mention of scopes, or bound variables. – NDB Jul 31 '23 at 13:25
  • When I see people writing $y_1$ and $y_2$ it makes a lot more sense but it is not standard notation, is that because most people don't worry about this issue much? – Nav Bhatthal Jul 31 '23 at 13:26
  • I have simply phrased these concepts differently. I am not sure that an extended discussion on these concepts shed more light here. @NDB – nicomezi Jul 31 '23 at 13:27
  • @NavBhatthal Yes, like in many languages, people agree to understand some ideas that are not explicitly said. Things get omitted. If someone doesn't understand, they ask for clarification, and that's it. – NDB Jul 31 '23 at 13:32
  • Thanks NDB for your help – Nav Bhatthal Jul 31 '23 at 13:33

1 Answers1

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It's just the result of using abbreviated notation.

When you say "Plot the graph $y=x^2$", what you really mean is "Plot the set of points $\{(x,y)\in \mathbb R^2 : y = x^2\}$". The $x$ and the $y$ are dummy variables used to show the relationship between the coordinates of the typical point on the graph of that function. They have absolutely no meaning outside of the set notation in which they are used.

So it it perfectly reasonable to say "Plot the two sets of points $\{(x,y)\in \mathbb R^2 : y = f(x)\}$ and $\{(x,y)\in \mathbb R^2 : y = g(x)\}$" because the variables $x$ and $y$ are only used within the set notation to describe the points in the set.

The corresponding "abbreviated" phraseology would be to say "Plot the two graphs $y=f(x)$ and $y=g(x)$". Knowing what is really meant by these removes any confusion.

MPW
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  • How do we get around the issue of having $y$ $=$ to two different values at once? – Nav Bhatthal Jul 31 '23 at 13:31
  • @NavBhatthal : But it isn't true that $y$ equals two different values at once. $y$ is a placeholder that allows you to speak of the generic point instead of having to speak of each specific point individually. Instead of having to say "$4=2^2$ and $9=3^2$ and $100=(-10)^2$ and $\frac14=\left(\frac12\right)^2$ and...", you can indicate the generic relationship by saying "$y=x^2$ for each point $(x,y)$ on the graph", or more briefly, just "$y=x^2$" with it being understood that the relationship is meant to hold for precisely the points on the graph. – MPW Jul 31 '23 at 13:36
  • @NavBhatthal : And then you can also say, that if $x=2$, then $y=4$ (understanding the context $y=x^2$ for points on the graph). – MPW Jul 31 '23 at 13:38
  • I am saying that when graphing $f$ and $g$ on cartesian $xy$ grid, for some value of $x$, say $x_i$ we have two (or more if the functions are not well-defined) values of $y$, namely $f(x_i), g(x_i)$, does that not mean we have two values of $y$? IF $x = x_i$ then $y=f(x_i)$ AND $y=g(x_i)$. – Nav Bhatthal Jul 31 '23 at 13:39
  • Refer to the third paragraph of my answer. In that case, your $y$ can only have meaning when being used in the description of ONE graph. When you speak of the OTHER graph, you are using the dummy variable in a different context. They are NOT the same variable. – MPW Jul 31 '23 at 13:41
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    You are the first person to tell me about the idea that $y$ is not the same in both definitions but rather a place-holder, and now that I have embraced your idea the confusion is wearing away. Everyone else online told me that the $y's'$ are the same and it was very very confusing to say the least. Thanks! – Nav Bhatthal Jul 31 '23 at 13:44
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    @NavBhatthal Read about bound variables, in particular the case of variables bound in function expressions. Also read about the scope of variables – NDB Jul 31 '23 at 13:57
  • Thanks NBD, do you know why in the last $5$ or so years of me learning mathematics at highschool level I have never heard anyone mention scope or bound variables? Is it taught later on, or just assumed that its obvious? – Nav Bhatthal Jul 31 '23 at 14:01
  • @NavBhatthal Same as when toddlers don't learn to speak by learning grammar first. Instead some rules are learned by imitation and being corrected if they say something wrong. – NDB Jul 31 '23 at 14:06
  • That makes me slightly embarrassed to have asked this question. – Nav Bhatthal Jul 31 '23 at 14:07
  • @NavBhatthal : No reason for embarrassment. You asked what many many others didn’t understand but were afraid to ask. They should thank you. – MPW Jul 31 '23 at 14:37
  • One last thing, in my example where I used $x_i$ is it better to write, for $f$ when $x = x_i$, $y = f(x_i)$, then the same for $g$? Rather than writing $y= f(x_i)$ AND $g(x_i)$? – Nav Bhatthal Jul 31 '23 at 14:44
  • @NavBhatthal : If you're really talking about specific values of $x$ and/or $y$, you can distinguish them with subscripts or various decorations. For example, if you are evaluating functions $f$, $g$, and $h$ at the points $x_1$ and $x_2$, you could use (say) $y_i = f(x_i)$, and $\hat{y}_i = g(x_i)$, and $\dot{y}_i=h(x_i)$. – MPW Jul 31 '23 at 16:44
  • But my method is better than saying AND right? Thats what I want clarification on, thanks – Nav Bhatthal Jul 31 '23 at 16:48