Range (function)
The range of a function is the set of all actual output values produced when the function is applied to every element of its domain. Unlike the codomain (which is the set of all possible outputs), the range is the set of values that are actually achieved. Determining the range is essential in understanding a function's behaviour and is applied in signal processing, statistics, and optimisation problems.
Key Formula
Range(f) = { f(x) | x is in the domain of f }
LaTeX: \text{Range}(f) = \{ f(x) \mid x \in \text{Domain}(f) \}
| Symbol | Meaning | Unit |
|---|---|---|
| f | The function | mapping |
| x | An element of the domain | input value |
| f(x) | The corresponding output | output value |
Worked Example
Problem
Find the range of f(x) = x² − 4 for x ∈ ℝ.
Solution
Step 1: Recognise that x² ≥ 0 for all real x. — Therefore x² − 4 ≥ 0 − 4 = −4. Step 2: As x → ±∞, f(x) → +∞. So the function takes all values from −4 upward. Step 3: At x = 0, f(0) = −4 (minimum value). The function increases without bound. — Range = [−4, +∞)
Answer
Range of f(x) = x² − 4 is [−4, +∞), with minimum value −4 at x = 0.
Range vs Codomain: Key Distinctions
| Concept | Definition | Example f(x)=x², f: ℝ→ℝ | Notes |
|---|---|---|---|
| Codomain | Set B in f: A→B (all possible targets) | ℝ (all reals) | Declared with function |
| Range | Actual set of outputs achieved | [0, +∞) | Subset of codomain |
| Image of element | f(a) for a single a in A | f(3) = 9 | Single output value |
| Pre-image | All x with f(x) = y | f⁻¹(9) = {−3, 3} | May have multiple values |
Interactive Tools
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Related Terms
Function (mathematics)
A function is a relation that assigns to each element of a set (called the domain) exactly one element of another set (called the codomain). Denoted f: A → B, it is a rule that maps every input to a unique output without ambiguity. Functions are among the most central concepts in mathematics and appear in calculus, linear algebra, programming, and virtually every applied science.
Domain (function)
The domain of a function is the complete set of all possible input values (x-values) for which the function is defined and produces a valid output. For a function f: A → B, the domain is the set A. Identifying the domain is a critical first step in analysing any function, as inputs outside the domain can lead to undefined expressions such as division by zero or square roots of negative numbers.
Bijection
A bijection (or bijective function) is a function that is both injective (one-to-one) and surjective (onto), meaning every element of the domain maps to a unique element of the codomain, and every element of the codomain is mapped to by exactly one element of the domain. Bijections establish a perfect one-to-one correspondence between two sets and are fundamental in defining cardinality, invertible functions, and isomorphisms across mathematics.
From Old French "range" (row, rank, line), from "ranger" (to arrange). In mathematics, "range" in the context of functions became standard usage in the 20th century to describe the image set of a function.