Restricted Domain — Definition, Examples & Explained

Restricted Domain

The use of a domain for a function that is smaller than the function's domain of definition. Note: Restricted domains are commonly used to specify a one-to-one section of a function.

Graph of f(x) = x², x ≥ 0, showing a parabola starting at origin curving upward right, illustrating a restricted domain on [0, ∞).

See also

Restricted function

Key Formula

f\!:\, D_{\text{restricted}} \to \mathbb{R}, \quad D_{\text{restricted}} \subset D_{\text{original}}

Where:

$f$ = The function whose domain is being restricted
$D_{\text{restricted}}$ = The restricted domain — a chosen subset of the original domain
$D_{\text{original}}$ = The function's full domain of definition (all inputs for which the function is defined)
$\mathbb{R}$ = The set of real numbers (the codomain)

Worked Example

Problem: The function f(x) = x² has domain (-∞, ∞), so it is not one-to-one. Restrict its domain so that an inverse function exists, and find that inverse.

Step 1: Identify why the full function fails the horizontal line test. For f(x) = x², note that f(-3) = 9 and f(3) = 9. Two different inputs give the same output, so f is not one-to-one on (-∞, ∞).

f(-3) = 9 = f(3)

Step 2: Choose a restricted domain where each output is hit by exactly one input. The standard choice is x ≥ 0.

D_{\text{restricted}} = [0, \infty)

Step 3: Verify one-to-one behavior on the restricted domain. On [0, ∞), x² is strictly increasing, so every horizontal line crosses the graph at most once.

\text{If } 0 \le a < b, \text{ then } a^2 < b^2

Step 4: Find the inverse by swapping x and y and solving for y.

y = x^2 \;\Rightarrow\; x = y^2 \;\Rightarrow\; y = \sqrt{x}

Step 5: State the inverse function and its domain.

f^{-1}(x) = \sqrt{x}, \quad x \ge 0

Answer: By restricting f(x) = x² to the domain [0, ∞), the function becomes one-to-one, and its inverse is f⁻¹(x) = √x.

Another Example

This example uses a trigonometric function instead of a polynomial, showing how restricted domains are essential for defining all inverse trig functions.

Problem: Restrict the domain of g(x) = sin(x) so that it becomes one-to-one, and state the resulting inverse function.

Step 1: Observe that sin(x) is periodic with period 2π, so it repeats its values infinitely many times. It is not one-to-one on (-∞, ∞).

\sin(0) = 0 = \sin(\pi)

Step 2: Choose the standard restricted domain for sine: the interval [-π/2, π/2]. On this interval, sine increases from -1 to 1 and passes the horizontal line test.

D_{\text{restricted}} = \left[-\tfrac{\pi}{2},\; \tfrac{\pi}{2}\right]

Step 3: With this restriction, the inverse function is the arcsine (also written sin⁻¹). Its domain is [-1, 1] and its range is [-π/2, π/2].

g^{-1}(x) = \arcsin(x), \quad -1 \le x \le 1

Step 4: Verify: sin(π/6) = 1/2, and arcsin(1/2) = π/6. The restricted function and its inverse undo each other.

\arcsin\!\bigl(\sin(\pi/6)\bigr) = \pi/6

Answer: Restricting sin(x) to [-π/2, π/2] makes it one-to-one, producing the inverse function arcsin(x) with domain [-1, 1].

Frequently Asked Questions

Why do you need to restrict the domain of a function?

You restrict a domain primarily to make a function one-to-one so that an inverse function can exist. A function must pass the horizontal line test to have an inverse, and many common functions (like x², sin x, cos x) fail this test on their full domains. By limiting the inputs to a carefully chosen interval, you ensure each output corresponds to exactly one input.

What is the difference between a restricted domain and a domain of definition?

The domain of definition is the largest set of inputs for which a function produces real outputs — it is determined by the function's formula. A restricted domain is a deliberate, smaller subset of the domain of definition chosen for a specific purpose, such as creating an invertible function. For example, x² has a domain of definition of all real numbers, but a common restricted domain is [0, ∞).

How do you choose which restricted domain to use?

You choose a continuous interval on which the function is strictly increasing or strictly decreasing, which guarantees it is one-to-one. There may be multiple valid choices. For x², both [0, ∞) and (-∞, 0] work. By convention, textbooks and calculators use the restriction that includes positive values or is most natural for the context.

Restricted Domain vs. Domain of Definition

	Restricted Domain	Domain of Definition
Definition	A chosen subset of the full domain, deliberately smaller	The largest set of inputs for which the function is defined
Who determines it	You (or a convention) choose it for a purpose	The function's formula determines it
Purpose	Typically to make the function one-to-one or to model a real-world constraint	To describe all valid inputs
Example for f(x) = x²	[0, ∞) or (-∞, 0]	(-∞, ∞)

Why It Matters

Restricted domains appear every time you study inverse functions. The definitions of arcsin, arccos, and arctan all rely on restricting the domains of sin, cos, and tan to specific intervals. In calculus, restricted domains also arise when defining piecewise functions or ensuring a function is monotonic on an interval for applying theorems like the Intermediate Value Theorem.

Common Mistakes

Mistake: Restricting the domain to an interval where the function is still not one-to-one.

Correction: Always verify that on your chosen interval the function is strictly increasing or strictly decreasing. For example, restricting f(x) = x² to [-2, 3] does not work because f(-2) = 4 = f(2), so it still fails the horizontal line test.

Mistake: Confusing the restricted domain of the original function with the domain of the inverse function.

Correction: The restricted domain of f becomes the range of f⁻¹, while the range of f on that restricted domain becomes the domain of f⁻¹. For f(x) = x² on [0, ∞), the inverse f⁻¹(x) = √x has domain [0, ∞) and range [0, ∞).

Related Terms

Domain — The set of all allowable inputs to a function
Domain of Definition — The full natural domain before any restriction
Function — The rule being restricted to a smaller domain
One-to-One Function — The property a restriction aims to achieve
Restricted Function — The function that results from applying a domain restriction
Inverse Function — Exists only when the function is one-to-one
Horizontal Line Test — Visual test to check if restriction made function one-to-one