Domain and Range Calculator with Graph

What Is Domain and Range?

Every function has an input set and an output set. The domain is the set of all x values the function can take. The range is the set of all y values it can produce.

Think of a vending machine. The domain is the set of buttons you can press. The range is the set of snacks it can give you. If a button does not exist, you cannot press it. If a snack is sold out, it is not in the range.

For f(x) = x², the domain is all real numbers, because you can square any number. The range is y ≥ 0, because a square is never negative.

How to Find Domain and Range from a Graph

The graph shows both answers at a glance. For the domain, look at how far the graph stretches left and right along the x-axis. For the range, look at how far it stretches up and down along the y-axis.

A parabola that opens up has a domain of all real numbers and a range that starts at its vertex and goes up. A square root graph starts at one point and moves right and up, so both its domain and its range start at that point.

Watch for breaks. If the graph has a gap, a hole, or a jump, some x or y values are missing. A rational function with a vertical asymptote at x = 2 has no value at x = 2, so 2 is not in the domain. The graphing rational functions calculator shows those breaks clearly.

Domain and Range of Common Functions

These are the cases you will meet most often in class:

• Linear functions (y = mx + b): domain all real numbers, range all real numbers.
• Quadratic functions (y = ax² + bx + c): domain all real numbers. Range starts at the vertex and goes up (a > 0) or down (a < 0).
• Square root functions (y = √(x - 2)): domain x ≥ 2, range y ≥ 0.
• Rational functions (y = 1 / x): domain all x except where the denominator is zero. Range is all y except the horizontal asymptote value in some cases.
• Absolute value (y = |x|): domain all real numbers, range y ≥ 0.
• Exponential functions (y = 2^x): domain all real numbers, range y > 0.

Interval Notation and Set-Builder Notation

Domain and range are usually written in interval notation. All real numbers is (-∞, ∞). The set x ≥ 2 is [2, ∞). A square bracket means the endpoint is included; a round bracket means it is not. Infinity always gets a round bracket, because infinity is not a number you can reach.

For a function with a gap, use the union symbol ∪. The domain of f(x) = 1 / (x - 2) is (-∞, 2) ∪ (2, ∞). In set-builder notation, the same domain is {x | x ≠ 2}, which reads "all x such that x is not equal to 2".

Common Mistakes to Avoid

• Confusing domain and range. Domain is about x, range is about y. Say the words as you work: "x can be..." for domain, "y can be..." for range.
• Forgetting the denominator. In a fraction, the denominator can never be zero. Always check it first.
• Forgetting the square root rule. The inside of a square root must be zero or positive.
• Using the wrong bracket. A closed circle on the graph means a square bracket. An open circle means a round bracket.

Practice With Our Free Tool

Use the domain and range calculator above to check your homework. Enter the function, read the domain and range, and compare them with your answer. For piecewise functions with several rules, find the domain of each piece separately, then combine them. Everything is free, with no download and no sign-up.

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