The graph of a function is the set of all input–output pairs of a function. For a real-valued function of one real variable, these pairs are represented as points ((x,f(x))) in a coordinate plane. The resulting graph may be a curve, isolated points, or a more complicated set; it need not be continuous. Graphs provide a geometric representation of dependence between variables, complementing formulas and tables. (openstax.org)
Formal definition
For a function (f:A\to B), its graph is [ \Gamma_f={(a,f(a)):a\in A}. ] Each ordered pair records an input and its unique output. Thus (\Gamma_f) is a subset of the Cartesian product (A\times B), and can be viewed as a binary relation satisfying the defining condition for a function: every element of (A) appears as the first component of exactly one pair. This set description expresses the same input–output correspondence represented by tables and plotted points. (openstax.org)
The domain is recovered from the graph as its set of first components. Its range is the set of second components that actually occur. The codomain (B), however, can contain elements outside the range and is not determined by the ordered pairs alone. The graph therefore records all function values without necessarily recording every part of the specification (f:A\to B). (openstax.org)
Coordinate representation and graphical tests
When inputs and outputs are real numbers, the graph is conventionally drawn with inputs on the horizontal axis and outputs on the vertical axis. This representation belongs to analytic geometry. It differs from the vertices-and-edges structures studied in graph theory, despite the shared word “graph.” (mathworld.wolfram.com)
The vertical line test characterizes function graphs in the plane: every vertical line must intersect the set in at most one point. A line (x=c) intersects it exactly once when (c) belongs to the domain, and not at all otherwise. A circle fails this test as a graph of (y=f(x)), since some inputs correspond to two heights. (openstax.org)
The horizontal line test addresses a different property. A function is injective precisely when every horizontal line intersects its graph at most once. Its inverse, defined on its range, then has a graph obtained by exchanging coordinates—geometrically, reflecting the original graph across (y=x). (openstax.org)
Reading information from a graph
A graph displays function values, domain restrictions, and intercepts. Its horizontal-axis intersections occur where (f(x)=0); its vertical-axis intersection is ((0,f(0))), provided zero belongs to the domain. Open and filled endpoint markers conventionally indicate exclusion and inclusion, respectively. (openstax.org)
For example, the polynomial function (f(x)=x^2), defined on all real numbers, has a parabolic graph and range ([0,\infty)). Restricting its domain to ([0,\infty)) retains only the right-hand half and makes the function injective. Domain restrictions therefore alter the graph even when the formula remains unchanged. (openstax.org)
Graphs also reveal intervals of increase or decrease, local extrema, symmetry, and discontinuities. A monotonic function maintains one direction of change over its domain or a specified interval. Such features describe the underlying function, not merely the appearance of a particular drawing. (openstax.org)
Transformations
Related functions produce systematically related graphs. For [ g(x)=a,f(b(x-h))+k,\qquad a,b\ne0, ] each original point ((u,f(u))) becomes [ \left(h+\frac{u}{b},,k+a f(u)\right). ] Consequently, (h) and (k) control translations, while (|a|) and (1/|b|) control vertical and horizontal scale factors. Negative (a) or (b) introduces a reflection. (openstax.org)
In particular, (f(x-h)) shifts a graph rightward by (h) when (h>0), whereas (f(x)+k) shifts it upward by (k). The graphs of (-f(x)) and (f(-x)) are reflections across the horizontal and vertical axes, respectively. These rules also transform domain restrictions and excluded points. (openstax.org)
Connections with calculus
In calculus, the derivative connects a function’s formula with its graph’s local behavior. Where (f) is differentiable, (f'(x)) gives the tangent slope. A positive derivative throughout an interval implies increase, and a negative derivative implies decrease. A critical point may be a local extremum, but need not be: (x^3) has zero derivative at the origin without a maximum or minimum there. (openstax.org)
The second derivative describes concavity: positive values indicate upward concavity, and negative values downward concavity. The definite integral gives signed area between a graph and the horizontal axis. For continuous functions (f\ge g) on ([a,b]), the area between their graphs is [ \int_a^b [f(x)-g(x)],dx. ] (openstax.org)
Higher-dimensional graphs and plotting limitations
For (f:D\subseteq\mathbb R^n\to\mathbb R), the graph consists of points ((x_1,\ldots,x_n,f(x_1,\ldots,x_n))) in ((n+1))-dimensional Euclidean space. A function of two variables is commonly represented by a surface in three dimensions. Its level sets, given by (f(x,y)=c), provide contour representations in the input plane. (openstax.org)
A displayed plot is not the exact graph: it shows a bounded viewing window with finite resolution. Numerical plotting samples function values and may miss narrow features or rapid oscillations. Adaptive sampling places additional points where variation is greatest, but graphical appearance alone cannot replace verification of a function’s properties. (mathworld.wolfram.com)