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Domain of Definition

Domain of Definition
Natural Domain

Alternate terms for domain used to make it clear that the domain being referred to is not a restricted domain.

 

 

 

See also

Range

Worked Example

Problem: Find the domain of definition of the function f(x)=x−3f(x) = \sqrt{x - 3}.
Step 1: Identify what restricts the input. A square root requires its argument to be greater than or equal to zero.
x−3≥0x - 3 \geq 0
Step 2: Solve the inequality for xx.
x≥3x \geq 3
Step 3: Express the domain of definition in interval notation.
[3,∞)[3, \infty)
Answer: The domain of definition of f(x)=x−3f(x) = \sqrt{x - 3} is [3,∞)[3, \infty). This is the largest possible set of real inputs for which the function is defined.

Another Example

Problem: Find the domain of definition of g(x)=1x2−4g(x) = \dfrac{1}{x^2 - 4}.
Step 1: Identify what restricts the input. A fraction is undefined when its denominator equals zero.
x2−4=0x^2 - 4 = 0
Step 2: Solve for the excluded values by factoring.
x2−4=(x−2)(x+2)=0  ⟹  x=2 or x=−2x^2 - 4 = (x-2)(x+2) = 0 \implies x = 2 \text{ or } x = -2
Step 3: The domain of definition is all real numbers except these two values.
(−∞,−2)∪(−2,2)∪(2,∞)(-\infty, -2) \cup (-2, 2) \cup (2, \infty)
Answer: The domain of definition of g(x)=1x2−4g(x) = \dfrac{1}{x^2 - 4} is all real numbers except x=−2x = -2 and x=2x = 2.

Frequently Asked Questions

What is the difference between the domain of definition and the domain of a function?
They often mean the same thing, but 'domain of definition' (or 'natural domain') specifically emphasizes the largest set of inputs for which the function is mathematically defined. The word 'domain' by itself can sometimes refer to a restricted domain — a smaller set chosen for a particular purpose, such as making a function one-to-one. Using 'domain of definition' removes that ambiguity.
How do you find the domain of definition of a function?
Look for operations that impose restrictions on the input: division (denominator cannot be zero), even roots (radicand must be non-negative), and logarithms (argument must be positive). Exclude any input values that violate these rules. Whatever remains is the domain of definition.

Domain of Definition (Natural Domain) vs. Restricted Domain

The domain of definition is the full set of inputs where the function is mathematically valid. A restricted domain is a deliberate subset of the domain of definition, chosen for a specific reason — for example, restricting f(x)=x2f(x) = x^2 to x≥0x \geq 0 so that it has an inverse. The domain of definition of x2x^2 is all real numbers, but its restricted domain might be [0,∞)[0, \infty).

Why It Matters

Knowing the domain of definition tells you exactly where a function is valid before you graph it, evaluate it, or combine it with other functions. When you compose two functions, the domain of the composition depends on the domains of definition of both. In applied problems, the natural domain also reveals physical constraints — for instance, a square-root model for distance only makes sense for inputs that keep the radicand non-negative.

Common Mistakes

Mistake: Assuming every function has domain (−∞,∞)(-\infty, \infty).
Correction: Many functions have natural restrictions. Always check for division by zero, even roots of negative numbers, and logarithms of non-positive numbers before stating the domain.
Mistake: Confusing the domain of definition with a restricted domain given in a problem.
Correction: If a problem says f(x)=x2,  x≥0f(x) = x^2, \; x \geq 0, the domain of definition of x2x^2 is all real numbers, but the function as stated uses the restricted domain [0,∞)[0, \infty). Read carefully to see which is being asked for.

Related Terms

  • Domain — General term for a function's input set
  • Restricted Domain — A deliberately limited subset of the domain
  • Range — The set of all output values of a function
  • Function — A rule assigning each input exactly one output
  • Interval Notation — Common way to express domains as intervals
  • Codomain — The set a function's outputs are drawn from