How To Find Domain In Interval Notation

How to Find Domain in Interval Notation

Understanding the domain of a function is a fundamental skill in algebra and calculus. The domain tells you all the possible input values (usually x) for which the function produces a real output. Expressing this set in interval notation provides a clear, concise way to communicate the allowable x‑values. Below is a step‑by‑step guide that walks you through the process, highlights common function types, and offers practical examples to reinforce the concept Small thing, real impact. Less friction, more output..

Introduction to Domain and Interval Notation

The domain of a function f(x) is the set of all real numbers x that can be substituted into the function without causing an undefined operation (such as division by zero or taking the square root of a negative number). Interval notation uses brackets and parentheses to describe continuous sets of numbers:

• [a, b] includes both endpoints a and b (closed interval).
• (a, b) excludes both endpoints (open interval).
• [a, b) includes a but excludes b.
• (a, b] excludes a but includes b.
• (-∞, a) or (a, ∞) indicate unbounded intervals, where ∞ is never included, so a parenthesis is always used.

When the domain consists of multiple separate intervals, we join them with the union symbol ∪.

General Steps to Find the Domain

Follow these systematic steps for any algebraic expression:

1. Identify the type of function (polynomial, rational, radical, logarithmic, trigonometric, piecewise, etc.).
2. List all operations that impose restrictions:
   • Division → denominator cannot be zero.
   • Even‑root (square root, fourth root, …) → radicand must be ≥ 0.
   • Logarithm → argument must be > 0.
   • Tangent, secant, etc. → avoid points where the function is undefined (e.g., cos x = 0 for tan x).
3. Solve each restriction inequality or equation to find the forbidden x‑values.
4. Combine the allowed intervals, using union where necessary, and write the result in interval notation.
5. Double‑check by testing a few sample points from each interval to ensure they satisfy the original function.

Common Function Types and Their Domain Restrictions

1. Polynomial Functions

Polynomials (e.g., f(x) = 2x³ − 5x + 7) have no restrictions; their domain is all real numbers.

• Interval notation: (-∞, ∞)

2. Rational Functions

For a rational function f(x) = P(x)/Q(x), set the denominator Q(x) ≠ 0 and solve.

• Example: f(x) = (3x + 1)/(x² − 4)
  • Denominator x² − 4 = 0 → x = ±2 are excluded.
  • Domain: (-∞, −2) ∪ (−2, 2) ∪ (2, ∞)

3. Square‑Root (Even‑Root) Functions

For f(x) = √[g(x)] (or any even root), require g(x) ≥ 0.

• Example: f(x) = √(5 − 2x)
  • Solve 5 − 2x ≥ 0 → x ≤ 2.5
  • Domain: (-∞, 2.5]

4. Cube‑Root (Odd‑Root) Functions

Odd roots (cube root, fifth root, …) accept any real radicand, so there is no restriction from the root itself.

• Example: f(x) = ∛(x + 3) → domain (-∞, ∞)

5. Logarithmic Functions

For f(x) = log_b[g(x)] (any base b > 0, b ≠ 1), demand g(x) > 0.

• Example: f(x) = log(x − 1)
  • Solve x − 1 > 0 → x > 1
  • Domain: (1, ∞)

6. Trigonometric Functions

• Sine and cosine: domain (-∞, ∞).
• Tangent and secant: undefined where cosine = 0 → x ≠ π/2 + kπ (k ∈ ℤ).
• Cotangent and cosecant: undefined where sine = 0 → x ≠ kπ.

Express these exclusions as a union of open intervals between the asymptotes.

7. Piecewise Functions

Determine the domain of each piece separately, then take the union of all pieces’ domains.

Worked Examples

Example 1: Mixed Rational and Radical

Find the domain of f(x) = √(x + 4) / (x² − 9).

1. Radical restriction: x + 4 ≥ 0 → x ≥ −4.
2. Denominator restriction: x² − 9 ≠ 0 → x ≠ ±3.
3. Combine: start with [−4, ∞) then remove −3 and 3.
   • Intervals: [−4, −3) ∪ (−3, 3) ∪ (3, ∞).

Example 2: Logarithmic with Quadratic Argument

Find the domain of f(x) = ln(x² − 5x + 6).

1. Argument must be > 0: x² − 5x + 6 > 0.
2. Factor: (x − 2)(x − 3) > 0.
3. Test intervals:
   • (-∞, 2): positive → keep.
   • (2