How to find domain and range?

Understanding the domain and range of a function is a fundamental aspect of algebra and calculus. The domain refers to the set of all possible input values (x-values) for which the function is defined, while the range refers to the set of all possible output values (y-values) the function can produce.

Finding the Domain

Identify any restrictions on the input values. For example, denominators cannot be zero, and square roots must have non-negative radicands.
Consider the type of function. Polynomial functions have a domain of all real numbers, while rational functions and functions with square roots have more limited domains.

Finding the Range

Understand the behavior of the function. For example, consider the highest and lowest values the function can take and how the function behaves as x approaches positive or negative infinity.
Use calculus, if necessary, to find critical points that can help determine the range.

Example: Domain and Range

Example 1: Consider f(x) = 1 / (x – 2).

Domain: Since the denominator cannot be zero, x cannot be 2. So, the domain is all real numbers except x = 2.

Range: As x approaches 2 from either direction, f(x) grows without bound, and as x approaches positive or negative infinity, f(x) approaches 0. Therefore, the range is all real numbers except y = 0.

Example 2: Consider f(x) = x².

Domain: As a polynomial, the domain of f(x) is all real numbers.

Range: Since x² is always non-negative, the range is all real numbers greater than or equal to 0.