Domain of a function

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable which will make the function work and will output real numbers. In simpler terms, it is the set of all possible inputs for the function that result in a defined output.

Understanding the Domain

When we define a function f(x), we are saying that f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E. The set D is the domain of the function.

Why is the Domain Important?

The domain is important because it specifies the values for which the function is defined. If we use values outside the domain, the function might not give a real number, or it might not make sense at all.

Determining the Domain

To find the domain of a function, we must identify all the values that the independent variable can take without leading to any undefined or non-real number.

Common Situations Affecting the Domain:

  1. Division by Zero: If the function has a denominator, the values that make the denominator zero are not included in the domain.
  2. Square Roots: If the function involves a square root, the values inside the square root must be greater than or equal to zero.
  3. Logarithms: If the function involves a logarithm, the argument of the logarithm must be greater than zero.

Table of Situations and Their Domain Restrictions

Situation Restriction Domain
Division by zero Denominator ≠ 0 All real numbers except those that make the denominator zero
Square roots Radicand ≥ 0 All real numbers that make the radicand non-negative
Logarithms Argument > 0 All positive real numbers

Formulas Involving the Domain

The domain can be expressed using inequality notation, interval notation, or set-builder notation. Here are some examples:

  • Inequality Notation: x > 0
  • Interval Notation: (0, ∞)
  • Set-Builder Notation: {x | x > 0}

Examples

Let's look at some examples to understand how to find the domain of different types of functions.

Example 1: Polynomial Function

Consider the function f(x) = x^2 - 5x + 6. Since there are no restrictions like division by zero or square roots, the domain is all real numbers.

  • Domain: (-∞, ∞) or R (the set of all real numbers)

Example 2: Rational Function

Consider the function f(x) = 1 / (x - 3). Here, the denominator cannot be zero, so x cannot be 3.

  • Domain: (-∞, 3) U (3, ∞)

Example 3: Square Root Function

Consider the function f(x) = √(4 - x). The expression under the square root must be non-negative, so 4 - x ≥ 0.

  • Domain: (-∞, 4]

Example 4: Logarithmic Function

Consider the function f(x) = log(x - 1). The argument of the logarithm must be positive, so x - 1 > 0.

  • Domain: (1, ∞)

Practice Problems

  1. Find the domain of f(x) = x / (x^2 - 1).
  2. Find the domain of f(x) = √(x + 3).
  3. Find the domain of f(x) = log(5 - x).

Solutions

  1. The denominator x^2 - 1 cannot be zero, so x cannot be 1 or -1. Domain: (-∞, -1) U (-1, 1) U (1, ∞)
  2. The expression under the square root must be non-negative, so x + 3 ≥ 0. Domain: [-3, ∞)
  3. The argument of the logarithm must be positive, so 5 - x > 0, which means x < 5. Domain: (-∞, 5)

In conclusion, understanding the domain of a function is crucial for correctly using and graphing functions. It ensures that we only consider the input values for which the function is defined and can produce real number outputs.