What exactly is the definition of a graph?

Question

The definitions I looked up online are very abstract and unspecific to my example below.

May I ask if the image I attached below contains one line graph? Or does it contain two line graphs (one represented by the orange line and another represented by the blue line)?

Thank you very much for your time.

Answer 1

I would say your image is one graph containing two curves.

Answer 2

Graph can mean several different things.

That is either

1) a graph (meaning graphical representation) containing two curves;

2) two graphs of two functions (where the graph of a function $f:A\to B$ is a subset $\Gamma_f$ of the Cartesian product $A\times B$ satisfying that for every $a\in A$ there is one and only one $b\in B$ such that $(a,b)\in\Gamma_f$).

So... Both, depending on what you mean by graph: graphical representation or graph of a function.

Answer 3

The word graph has many meanings, both in mathematics and in vernacular English. In this particular case, I think that the confusion is caused by a collision between a mathematical meaning of the word, and a non-mathematical meaning.

• Everyday English Meaning: As per Google, a graph is

  a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.

  This definition asserts that a graph is a way of displaying data. Examples of such graphs might include the sorts of things which mathematicians think of as graphs, but can also include bar graphs, pie charts, scatterplots, and so on.

• Mathematical Meaning: In this context, the relevant mathematical definition of graph is something like the following: if $f : X \to Y$ is a function, then the graph of $f$ is the set of ordered pairs of the form $(x,f(x))$, where $x$ is any point in $X$, and $f(x)$ is the image of $x$ in $Y$. While this definition is pretty technical, it is essentially the definition used in most high school algebra classes, and is therefore likely to be one of the first uses of the word encountered by most students.

  By way of example, the following image shows the graphs of the function (from left to right) $f(x) = x^2$, $g(x) = \mathrm{e}^x$, and $h(x) = \tan(x)$.

  

  Per this definition, the image in the question could be said to depict two curves, each of which is the graph of some function. It would not be entirely wrong to say that the image "is" two graphs, though I think that this verb is a bit awkward. It would be better to say that the image "shows" or "depicts" or "displays" two graphs (or, even better, "the image shows the graphs of two functions").

It may also be worth noting that the word "graph" has yet more meanings, both in English and in mathematics. For example, a graph (in the sense of graph theory) is a collection of nodes or vertices which are joined by edges; while a graph in English (or, perhaps, in linguistics) refers to a written symbol which represents a phoneme, word, or other piece of language.

However, these other meanings really aren't relevant here.