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For questions regarding the plotting or graphing of functions. For questions about the kinds of graphs with vertices and edges, use the (graph-theory) tag instead.

Given a real-valued function $$f\colon \mathbf{R} \to \mathbf{R}$$, the graph of $$f$$ is the set of all input-output pairs $$(x,f(x))$$ regarded as a set of points in the plane $$\mathbf{R} \times \mathbf{R}$$. Considering the graph of a function gives us a geometric perspective on the data that the function represents.

• If the function $$f$$ is continuous, the graph of $$f$$ "looks continuous." That is, there are no gaps, and the graph is a connected curve.

• If the function $$f$$ is differentiable, then it will contain no "sharp corners."

• If we're thinking of the domain of the function as representing time, the the graph gives us a nice visualization of the change in outputs of the function over time.

A graph can be defined much more generally though. Let $$\mathbf{k}$$ be a local field, and suppose $$f$$ is a vector-valued function $$f\colon \mathbf{k}^n \to \mathbf{k}^m$$ where $$f(x_1, \dotsc, x_n) = (y_1, \dotsc, y_m)$$ and each coordinate $$y_i$$ of the output is a function of the $$x_1, \dotsc, x_n$$. In this setting, the graph of $$f$$ is the set of points

$$(x_1, \dotsc, x_n, y_1, \dotsc, y_m) \subset \mathbf{k}^{n+m}\,.$$

This general construction of the graph of a function can be useful in the study of algebraic geometry or the study of manifolds.