The domain is the set of input values for which a function is defined. The range is the set of output values the function actually takes. A graph makes both sets visible, provided you read endpoints, gaps, and asymptotes carefully.
Read the domain horizontally
Imagine moving from left to right across the graph. Record every x-value that has at least one point on the curve. Empty vertical slices indicate excluded x-values.
Read the range vertically
Now move from bottom to top. Record every y-value reached by at least one point on the curve. Empty horizontal levels are excluded from the range.
Example 1: a square-root function
Graph f(x)=sqrt(x-2). The curve begins at (2,0) and continues to the right.
- Domain:
[2,∞) - Range:
[0,∞)
The endpoint is included because the function is defined at x=2 and the graph contains the point (2,0).
Example 2: a rational function
Graph f(x)=1/(x-1). The graph has a vertical asymptote at x=1 and a horizontal asymptote at y=0.
- Domain: all real numbers except
1 - Range: all real numbers except
0
The curve approaches the asymptotes but never touches them.
Example 3: a quadratic function
For f(x)=x^2-4x+3, the parabola extends indefinitely to the left and right, so its domain is all real numbers. Its minimum value is -1 at the vertex (2,-1), so the range is [-1,∞).
Open and closed endpoints
On a manually defined interval, a filled point normally indicates inclusion and an open point indicates exclusion. In interval notation, use square brackets for included finite endpoints and parentheses for excluded endpoints. Infinity always uses a parenthesis.
Gaps, holes, and discontinuities
A missing point can remove one x-value from the domain, one y-value from the range, or both. For example, (x^2-1)/(x-1) simplifies to x+1 except at x=1. Its graph is the line y=x+1 with a hole at (1,2).
Do not rely only on the visible window
A finite screen cannot display infinity. Before deciding that a graph stops, consider whether it simply leaves the viewport. Zoom out, inspect the formula, and use limits or algebra when needed.
A reliable checklist
- Identify restrictions from denominators, even roots, and logarithms.
- Look for endpoints and whether they are included.
- Locate vertical asymptotes and holes.
- Find minimum or maximum y-values.
- Check whether the graph continues beyond the window.
- Confirm the result algebraically when an exact answer is required.
Open the Graphing Calculator and compare these examples →
Next, learn how to identify x-intercepts, y-intercepts, and intersections.