Can two different functions have the same graph?

Question

I know that, Identical Functions (Equal Functions) are those functions which have the same domain and give the same output for every input value. These functions have the same graph.

For example,

The functions $f(x)=x^3/x$ and $g(x)=x^4/x^2$ have the same domain (set of non-zero real numbers), and give the same output for every input value. These are identical functions (equal functions) and generate the same graph. On the other hand, the function $h(x)=x^2$ is not identical to the functions $f$ and $g$, as the domain of $h$ (set of real numbers) is different from that of $f$ and $g$ (set of non-zero real numbers). The only difference in the graphs of $h$ and $f$(or $g$) is at the point $x=0$.

Now coming to my doubt,

Can two different functions have the same graph? Or in other words, if the graphs of two different functions are exactly same, can we conclude that the two functions are identical (equal) ? If not please give some examples where two different functions generate the same graph.

Answer 1

No, Same graph $:=$ Same function. But there's a distinction about functions as rules rather than as graphs, which I think is what's leading to your confusion...

Functions as rules refers to the procedure used to go from an argument to a value, and this is the older notion of "function". That functions can also be considered as graphs, i.e., as sets of $(argument,value)$ pairs, is a later idea usually attributed to Dirichlet.

So your $x^3/x$ versus $x^4/x^2$ just illustrates two different procedures leading to the same graph. And then $x^2$ is yet another different procedure, whose graph moreover contains a $(0,0)$ element that the first two presumably don't. So it's indeed a (slightly) different function either way you look at it.

Answer 2

Yes... and no, depending on what you mean by "graph", and what you call identical functions. In other words, in order for your quesiton to have a precise answer, you must define "graph" precisely.

Here's an example of what I mean: if you mean something like plotting $(x,f(x))$ for all $x$ in the domain and seeing the graphical output (on, say, Geogebra or Desmos), then certainly it is possible for two different functions to appear the same. Your example is excellent, the map $x\mapsto x^3/x$ is not defined at $0$, but if you drew the "graph" and looked at the graphical output, this one point is hardly distinguishable. Also, the graph of say $x$ and $\lfloor Nx\rfloor/N$ will look the same for sufficiently large $N$.

On the other hand, if you were more rigorous and defined the graph of a function, $G(f)$ to be the set $$G(f):=\{(x,f(x))\mid x\in D_f\}$$ where $D_f$ is the domain of $f$, then the answer is no: if $G(f)=G(g)$ then $f=g$. First the domains of $f,g$ must coincide, since if there were a point $x\in D_f$ but $x\notin D_g$, then $(x,f(x))$ is in $G(f)$ but not $G(g)$, contradicting the assumption. On the other hand for each $x$ by hypothesis $(x,f(x))=(x,g(x))$, then $f(x)=g(x)$ for all $x$. This is exactly what one means by an identical function.