Mathwords logoMathwords

Range of a Function — Definition, Formula & Examples

The range of a function is the set of all possible output values (y-values) the function can produce. If you plug every value from the domain into the function, the collection of results you get is the range.

Given a function f:A→Bf: A \to B, the range of ff is the set {f(x)∣x∈A}\{f(x) \mid x \in A\}, that is, the subset of the codomain BB consisting of all elements that are actually mapped to by at least one element of the domain AA.

How It Works

To find the range, ask: what y-values can this function actually reach? For a simple function like f(x)=x2f(x) = x^2, notice that squaring any real number always gives zero or a positive result, so the range is [0,∞)[0, \infty). For linear functions like f(x)=3x+1f(x) = 3x + 1, every real y-value is reachable, so the range is all real numbers. When working with graphs, the range is the set of y-values covered by the curve — scan from bottom to top and note which heights the graph reaches.

Worked Example

Problem: Find the range of f(x)=−2x2+8f(x) = -2x^2 + 8.
Identify the shape: This is a downward-opening parabola (the coefficient of x2x^2 is negative), so it has a maximum value at its vertex.
Find the vertex: The vertex occurs at x=−b2ax = -\frac{b}{2a}. Here a=−2a = -2 and b=0b = 0, so the vertex is at x=0x = 0.
f(0)=−2(0)2+8=8f(0) = -2(0)^2 + 8 = 8
Determine the range: Since the parabola opens downward, the y-values go from 88 down to −∞-\infty. No output can exceed 8.
Range=(−∞, 8]\text{Range} = (-\infty,\, 8]
Answer: The range is (−∞,8](-\infty, 8], meaning all real numbers less than or equal to 8.

Why It Matters

Finding the range is essential whenever you need to know the limits of a function's output — for instance, determining the maximum profit in a business model or the peak height of a projectile. It appears repeatedly in Algebra 2, Precalculus, and Calculus courses when analyzing and graphing functions.

Common Mistakes

Mistake: Confusing the range with the codomain or with the domain.
Correction: The domain is the set of valid inputs (x-values). The codomain is the set where outputs are allowed to land, but the range is only the outputs that actually occur. For f(x)=x2f(x) = x^2 with codomain R\mathbb{R}, the range is [0,∞)[0, \infty), not all of R\mathbb{R}.