Question

what is the domain of a function and how can you find it from a graph

Answer 1

Domain: the set of all possible input x-values for which the function is defined (i.e., the x-values that actually produce a point on the graph).

How to find the domain from a graph — quick steps
1. Project the graph onto the x-axis: look at the horizontal extent of the curve(s).
2. Include x-values where the graph has a point. If the graph keeps going left or right with arrows, include ±∞ as appropriate.
3. Watch endpoints and special marks:
- Filled (solid) dot at x = a → include a.
- Open dot at x = a (a hole) → exclude a.
- Vertical asymptote at x = a (graph goes to ±∞) → exclude a.
4. Combine the included x-intervals and express the domain in interval notation or set-builder form.

Common examples
- A parabola or line that extends forever left and right → domain = (-∞, ∞).
- Graph starts at x = 0 with a filled dot and continues right → domain = [0, ∞).
- Rational function with a vertical asymptote at x = 2 → domain = (-∞, 2) ∪ (2, ∞).
- A graph with a hole at x = 1 (open dot) but otherwise continuous → domain excludes 1: (-∞, 1) ∪ (1, ∞).

Tip: If you have the formula instead of the graph, find x-values that make the formula undefined (division by zero, even root of negative, log of nonpositive) to get the domain; the graph method is simply reading those exclusions visually.