What Is Range of a Function?
The range of a function is the set of all output values. To find it, look for the smallest and largest outputs on the domain, using the vertex, the endpoints and any asymptotes.
Choose a function type and enter its coefficients. You can restrict the domain to see how it changes the range. The calculator gives the range and the domain used, the minimum and maximum, the vertex where it applies, and a graph.
Below are the key rules, a worked quadratic example and answers on domain versus range.
How to Use the Range of a Function Calculator
- Pick the function type.
- Enter the coefficients.
- Optionally restrict the domain.
- Read the range and the graph.
Range of a Function Formula
Vertex of a quadratic
Range of a Function Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| a, b, c | Coefficients of the function | |
| h, k | Shift in x and y | |
| Vertex | Turning point of a parabola or V shape | |
| Domain | Allowed inputs |
Range of a Function Worked Example
For f(x) = x² − 4x + 3, a = 1, b = −4. The vertex is at x = 4 ÷ 2 = 2 and f(2) = 4 − 8 + 3 = −1. Since a > 0 the parabola opens upward, so the range is [−1, ∞).
With the example values loaded in the calculator above, the steps are:
- Function: f(x) = x² − 4x + 3.
- Domain used: (−∞, ∞).
- Vertex at x = 2, f = -1.
- The smallest value is -1 and the largest is ∞; the function is continuous, so every value between them is reached.
- Range = [-1, ∞).
Result: Range = [-1, ∞); Domain used = (−∞, ∞).
How to Read the Range
The range is the set of output values a function can take. For f(x) = x² − 4x + 3 the vertex is at x = 2 and f(2) = −1. Because the parabola opens upward, the lowest output is −1 and the range is [−1, ∞).
Restricting the domain changes the range. If you only allow x between 0 and 1, the outputs fall between f(1) = 0 and f(0) = 3, which is a much smaller set. The calculator shows the domain it used so you can check it.
Where Range of a Function Is Used
Quadratics
Find the vertex and the resulting minimum or maximum value.
Absolute value functions
See that the range starts at the vertex value and goes up or down from it.
Restricted domains
Find the range when x is limited to an interval.
Homework
Check range answers written in interval notation.
Common Range of a Function Mistakes to Avoid
- Domain is the set of allowed inputs. Range is the set of outputs. Do not swap them.
- For a quadratic, the range starts or stops at the vertex value. The sign of a decides which.
- A restricted domain can cut the range off before the vertex.
- Write the answer in interval notation and use a square bracket for a value that is reached.
Range of a Function Terms and Definitions
- Range
- The set of all output values f(x).
Range of a Function FAQs
How do you find the range of a function?
Work out what outputs are possible. For a quadratic, find the vertex and check which way it opens. For a linear function with no restriction, the range is all real numbers.
What is the range of a quadratic?
For f(x) = ax² + bx + c, the vertex value is f(−b ÷ 2a). If a > 0 the range is [vertex value, ∞). If a < 0 it is (−∞, vertex value].
What is the range of an absolute value function?
For a|x − h| + k, the vertex value is k. If a > 0 the range is [k, ∞), and if a < 0 it is (−∞, k].
What is the difference between domain and range?
The domain is every input x the function accepts. The range is every output f(x) it can produce.