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.
Best Answer
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.