We know an equation when plotted on a graph is a representation of a function if the graph passes the vertical line test.
Consider $x=y^2$. Its graph is a parabola and it fails the vertical line test. If we calculate $y$ from the above, we get $y = \pm\sqrt{x}$. That is, for each $y$ there are two $x$s.
Why does it matter if a graph is a function?
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$\begingroup$Functions and Graphs
A (real valued) function (of real variable) is a rule that assigns to every valid input (which is a real number) one and only one output (which is also a real number).
There are many ways to specify a function. One way is via a formula that tells you to take the input, perform some operations on it, and get the output. $f(x) = x^2$ is one such; it tells you to take the input given, $x$, square it, and whatever you get is the output. So if the input is $2$, the output is $2^2 = 4$. Etc.
Another way to specify a function, when there are only finitely many valid inputs, is via a table: you list all possible inputs, and what the corresponding output is. An example of this would be a grade sheet, where the input is the student ID number, and the output is the corresponding student’s GPA.
When you have infinitely many inputs, you can’t do a list. But there is a way to do it which is very similar, and that is a graph. If you have a graph (a collection of points on the plane), each point on the graph has associated a pair of coordinates, $\mathbf{p}=(a,b)$. You interpret the point on the graph as telling you that the input $a$ yields the output $b$.
Not every graph corresponds to a function, because if you have two different points with the same first coordinate, you won’t know what the output is (remember there can be only one output to each given input; if you have points $(4,2)$ and $(4,-2)$ in your graph, you don’t know if the output to $4$ is $2$ or $-2$). This is the so-called ”vertical line test”. For a graph to be usable to define a function, it must satisfy that every vertical line intersects the graph at most once; that is, there is at most one point on the graph with any give first coordinate.
Similarly, every (real valued) function (of real variable) defines a “graph”, which is kind of like the whole table of values of the function: namely, the graph of the function $y=f(x)$ is the collection of all points $\mathbf{p}=(a,b)$ for which $b$ is the output of the input $a$,
So every function gives you a graph, but not every graph gives you a function; some do, and some do not.
Equations and Graphs
An equation is simply an expression with an equal sign, asserting that two things are equal (that’s where the “equa” of “equation” comes from; an equation must have an equal sign). If you have an equation involving two variables, such as “$x=y^2$”, then it can be used to define a “graph” on the plane; namely, the graph of the equation is the collection of all points $\mathbf{p}=(a,b)$ such that, if you substitute $a$ for $x$ and $b$ for $y$ into your equation, you get a true statement. For the equation $x=y^2$, the point $(4,2)$ is in the graph, because if you substitute you get “$4=2^2$”, which is true; but $(0,1)$ is not in the graph of the equation because if you substitute you get “$0=1^2$”, which is false.
So every equation of two variables defines a graph, just like every function defines a graph.
But remember that while some graphs define a function, not every graph defines a function. Similarly, some graphs may be given by equations, but not every graph can be given by an equation (involving a certain type of operations).
Connection
If you have an equation, it defines a graph. That graph may or may not be the graph of a function (that graph may or may not be used to define a function) depending on whether it passes the horizontal line test or not. It’s possible that an equation will define a function that is not obvious from the equation.
Every function gives you a graph, and this graph always gives you an equation; namely, if the function is $f(x)$, then the equation $y=f(x)$ defines the same graph as the function you started with.
But not every equation gives you a function.
So there is a connection, but there is not a perfect connection:
- Every function gives you a graph.
- Not every graph gives you a function.
- Every equation gives you a graph.
- Not every graph gives you an equation.
- Every function gives you a graph that gives you an equation.
- Not every equation gives you a graph that gives you a function.
When an equation gives you a graph that gives you a function, sometimes we say that the equation “defines a function”, either explicitly, and sometimes implicitly. It does so via the following process: given an input, substitute it into the equation, and let the output be the (only) value that you can substitute into the equation to make it true. If, after you substitute, the value is immediately apparent, we say it is an explicit definition. For example, $y=x^2$ is an equation that gives an explicit definition: plug in an input, like $x=3$, and you get $y=3^2=9$ which immediately gives you the answer. Other times after substituting you need to do some work to figure out the value of the output; then we say the definition is implicit. For example, $yx-y = 3-yx$ defines a function implicitly. Given, e.g., $x=3$, you get $3y-y = 3-3y$, so then you “solve for $y$” to get $5y = 3$ or $y=\frac{3}{5}$. As it happens, this equation can be shown to give a graph that does define a function, so the equation definitely defines a function as well.
In your example, $x=y^2$, you have an equation that defines a graph that does **not* define a function (it fails the vertical line test, as you note).
On the other hand, the equation $y=x^2$ is an equation that defines a graph that does define a function (in fact, it defines it explicitly).
$\endgroup$ $\begingroup$That's a good question. But it's a little bit context sensitive: if you asked "Does it matter if I catch the bus on time?" my answer might be "Not to me!", while you might think it was very important.
If you're interested in describing fairly general shapes, the way we do in differential geometry and topology, you find out pretty fast that most of them cannot be represented as the graphs of functions on something nice like the $x$-axis, or the $xy$-plane.
On the other hand, when things are representable as graphs, we often have a lot of other tools that let is describe them very carefully and precisely, tools that don't work as well (if at all) in the non-graph situation. So when we're looking at those "fairly general" shapes I talked about, we do things like say "if I just look at this small part of the shape (like, say, the upper arc of a circle), then it's actually a graph!", and we actually require that every "small enough" part of our shapes looks graph-like (from some direction).
Your rotated parabola isn't graph-like from the $x$-direction, but if you think of $x$ as a function of $y$, then it is graph-like in the other direction. THere are other shapes -- like a circle -- that aren't graph-like in any direction, but if you look at small pieces of them, are in fact graph-like.
One of the big theorems of calculus says that for certain easy-to-describe shapes, you can show -- with only a little work -- that they are graph-like near either a single point, or even near every point of the shape, and this "Implicit Function Theorem" can save us a lot of grief -- we say "I know that this thing is graph-like, so it has all these nice properties that I've proved earlier...but because the IFT tells me it's graph-like, I don't need to actually write down the function whose graph matches it!" Because writing down that function is often really, really difficult, this theorem is one of the sledgehammers of beginning calculus/analysis.
I'm sorry to be so vague, but I'm guessing at your level in mathematics from the way you phrased your question, and hoping I've aimed at about the right level.
Short answer: graph-like means it's easy to prove theorems about the shapes. Not-graph-like objects may be "locally graph-like" and therefore those theorems may still apply.
$\endgroup$ $\begingroup$Given a function $f(x)$ we define the graph of $f$ to be the set of points given by$$\{(x,y) \ | \ f(x) = y\}$$The "vertical line test" of a function simply means that for any $x$, there is a unique value $y$ such that $f(x) = y$. If a graph defined by a smooth curve fails the vertical line test, there is still an associated well defined function. Consider two parameterized functions $x(t)$ and $y(t)$. Then the set$$\{(x(t),y(t)) \ | \ t\in [0,1] \}$$which defines a curve in $\mathbb{R}^2$, may fail the vertical line test. However the the graph of $f(t) := (x(t),y(t))$ is given by$$\{ (t,x,y) \ | \ f(t) = (x,y) \}$$will pass this test (now in 2 dimensions). That is for any $t$, there is exactly one value of $f(t)$, namely $(x(t),y(t))$. So when we talk about a smooth curve in space, there will be some function which describes the curve. Then it just comes down to whether we care about properties of the function, the curve or the graph.
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