How do you find the domain from a graph?

Look at how far left and how far right the graph extends along the $x$-axis. The domain includes every $x$-value where the graph has a point. If the graph has a filled dot at an endpoint, that value is included; an open dot means it is excluded. For example, a graph that starts at $x = -3$ (filled dot) and continues to the right without end has domain $[-3, \infty)$.

Domain of a Function — Definition, Formula & Examples

The domain of a function is the complete set of input values (usually

x

-values) for which the function is defined and produces a real output.

Given a function $f: A \to B$ , the domain is the set $A$ consisting of all elements $x$ such that $f(x)$ exists in the codomain $B$ . When working with real-valued functions, the domain is the largest subset of $\mathbb{R}$ for which the rule $f(x)$ yields a real number, unless a different domain is explicitly stated.

How It Works

To find the domain, ask: "Which

x

-values would cause a problem?" The three most common restrictions for real-valued functions are division by zero, square roots (or even roots) of negative numbers, and logarithms of non-positive numbers. Exclude those problematic

x

-values, and everything that remains is the domain. For polynomial functions like

f(x) = 3x^2 - 5x + 1

, no operation is restricted, so the domain is all real numbers. When a function is defined by a graph, the domain is the set of all

x

-values that the graph covers from left to right.

Worked Example

Problem: Find the domain of

f(x) = \dfrac{\sqrt{x - 2}}{x - 5}

Identify restriction 1 — square root: The expression under the square root must be non-negative.

x - 2 \ge 0 \implies x \ge 2

Identify restriction 2 — denominator: The denominator cannot equal zero.

x - 5 \neq 0 \implies x \neq 5

Combine both restrictions: Take the values satisfying

x \ge 2

and remove

x = 5

\text{Domain} = [2,\, 5) \cup (5,\, \infty)

Answer: The domain is

[2, 5) \cup (5, \infty)

, meaning all real numbers from 2 onward except 5.

Another Example

Problem: Find the domain of

g(x) = \ln(4 - x)

Apply the logarithm restriction: The argument of a natural logarithm must be strictly positive.

4 - x > 0

Solve the inequality: Isolate

x

x < 4

State the domain: All real numbers less than 4.

\text{Domain} = (-\infty,\, 4)

Answer: The domain is

(-\infty, 4)

Why It Matters

Domain shows up constantly in Algebra 2, Precalculus, and AP Calculus when you analyze or compose functions. In calculus, you must know the domain before you can discuss limits, continuity, or differentiability. Engineers and data scientists also check domains to ensure their mathematical models only receive inputs that produce meaningful results.

Common Mistakes

Mistake: Forgetting to check multiple restrictions at once.

Correction: A function like

\dfrac{\sqrt{x}}{x - 3}

has two restrictions:

x \ge 0

from the square root AND

x \neq 3

from the denominator. You must combine both to get the correct domain

[0, 3) \cup (3, \infty)

Mistake: Assuming every function has domain all real numbers.

Correction: Only certain types — mainly polynomials, exponential functions, and sine/cosine — have domain

(-\infty, \infty)

. Always inspect the formula for division, even roots, and logarithms before concluding the domain is unrestricted.

Related Terms

Argument of a Function — The input value that must lie in the domain
Dependent Variable — Output variable whose values depend on domain inputs
Continuous Function — Continuity is only evaluated on the domain
Discontinuity — Points in the domain where continuity breaks
Bounded Function — Boundedness is defined over the function's domain
Absolute Value Function — Example of a function with domain all reals
Constant Function — Simplest function; domain is all real numbers
Ceiling Function — Piecewise-defined function with domain all reals