Yes, in the most fundamental mathematical sense, every function inherently has a graph. This is because, in set theory, the graph of a function is not just a drawing but is often considered to be the function itself—a specific collection of ordered pairs that defines the relationship.
The Fundamental Definition of a Graph
Mathematically, a function is a rule that assigns each input element from a set (the domain) to exactly one output element in another set (the codomain). The graph of a function $f$ is formally defined as the set of all ordered pairs $(x, f(x))$ where $x$ is an element of the domain of $f$.
- A function is a set of ordered pairs: From this perspective, the existence of a function directly implies the existence of its graph. If you can define a function, you have inherently defined a set of input-output pairs, which constitutes its graph.
- Beyond visualization: This definition does not require the graph to be visually representable on a two-dimensional plane. It simply states that a specific collection of points exists.
Distinguishing 'Having' from 'Sketching'
While every function has a graph as a set of points, not all functions can be easily or meaningfully sketched or visualized on a typical coordinate plane. The ability to sketch a graph often implies properties like continuity, smoothness, or at least a clear pattern of discrete points.
Consider the following distinction:
| Aspect | "Having" a Graph (Mathematical Definition) | "Sketching" a Graph (Visual Representation) |
|---|---|---|
| Definition | The function itself, as a set of ordered pairs | A visual drawing on a coordinate plane |
| Universality | Yes, every function inherently possesses one | No, not all functions can be effectively drawn |
| Primary Purpose | Defines the function's input-output relationships | Aids in understanding and analyzing function behavior |
| Complexity Handled | Can represent infinitely complex relationships | Limited by human perception and drawing tools |
An Example: The Dirichlet Function
A prime example of a function that has a graph but cannot be sketched is the Dirichlet function, often denoted as $D(x)$ or $\chi_{\mathbb{Q}}(x)$:
$D(x) = \begin{cases} 1 & \text{if } x \text{ is a rational number} \ 0 & \text{if } x \text{ is an irrational number} \end{cases}$
Let's break down why this function fits the description:
- It has a graph: For every real number $x$, the function assigns a value (either 1 or 0). Thus, it has a definite set of ordered pairs, such as $(0,1)$, $(\sqrt{2},0)$, $(0.5,1)$, $(\pi,0)$, etc. This collection of points is its graph.
- It cannot be sketched: If you try to sketch this function on a coordinate plane:
- Rational numbers are infinitely dense on the real number line, meaning between any two rational numbers, there are infinitely many irrational numbers, and vice versa.
- This implies that the graph would consist of an infinite number of points at $y=1$ and an infinite number of points at $y=0$, with these two sets of points being inextricably intertwined and impossible to separate visually.
- The "graph" would appear as two indistinguishable, dense lines across the x-axis, one at $y=0$ and one at $y=1$, making it impossible to truly visualize the distinct nature of the function's values for different types of numbers.
In conclusion, while the concept of a "graph" often brings to mind a visual drawing, its mathematical definition is much broader. Every function, by its very nature as a set of ordered pairs, fundamentally possesses a graph. However, the complexity of some functions means their graphs cannot be easily, or at all, translated into a comprehensible visual representation.
