Graph of a function
Mathematically, the graph of a function is the collection of all pairs (x, f(x)) of the function. In particular graph means the graphical representation of this collection, in the form of a curve, together with axes, etc. Graphing is sometimes referred to as curve sketching.
The graph of the function
The graph of the cubic polynomial on the real line
Therefore the graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis.
The concept of the graph of a function is generalised to the graph of a relation. Note that although a function is always identified with its graph, they are not the same because it will happen two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned above is a surjection if its codomain is the real numbers but it is not if its codomain is the complex field.