Domain of a Function

Meaning of the domain of a function, how to identify valid inputs, and simple examples to understand the idea.

1. What the Domain Means

The domain of a function is the set of all inputs for which the function is actually defined. These are the values you are allowed to plug into the function. If an input lies outside the domain, the function has no assigned output for it.

2. Why the Domain Matters

Not every rule works for every possible number or object. For some inputs, the rule may break or not make sense. The domain tells you where the function “works safely”.

3. Formal Definition

If a function \( f: A \to B \) is defined, then the domain is A — the set from which the inputs are taken.

3.1. Definition

\text{Domain}(f) = A

Every element of A must have exactly one output.

4. Finding the Domain from a Formula

When a function is given by a formula, the domain is all values for which the formula makes sense. Often, certain expressions restrict the domain.

4.1. Common Restrictions

  • Denominators cannot be zero.
  • Inside a square root, the expression must be ≥ 0 (for real functions).
  • Inside a logarithm, the expression must be > 0.
  • Inside trigonometric inverses, inputs must lie in valid ranges.

5. Examples of Domains

Here are some common functions and their domains:

5.1. Example 1

\( f(x) = x^2 \)

No restrictions. Any real number works.

\text{Domain} = \mathbb{R}

5.2. Example 2

\( f(x) = \frac{1}{x} \)

x cannot be zero.

\text{Domain} = \mathbb{R} \setminus \{0\}

5.3. Example 3

\( f(x) = \sqrt{x - 3} \)

Expression inside square root must be ≥ 0.

x - 3 \ge 0 \Rightarrow x \ge 3

\text{Domain} = [3, \infty)

5.4. Example 4

\( f(x) = \ln(x) \)

Logarithm only works for positive inputs.

\text{Domain} = (0, \infty)

6. Domain from a Set of Ordered Pairs

When a function is written as ordered pairs, the domain is simply the set of all first elements.

6.1. Example

f = \{(1,3),(2,4),(5,6)\}

Domain = {1, 2, 5}

7. Choosing a Domain Intentionally

Sometimes we restrict a domain on purpose. This often happens when we want the function to behave in a specific way, like making an inverse possible.

7.1. Example

\( f(x) = x^2 \)

This function is not one-one on all real numbers, but if we restrict the domain to x ≥ 0, it becomes one-one.