Domain and range usually come up for the first time in Algebra 1 or 2, right around when functions are introduced. Most students get through the initial explanation, follow along in class, and then find themselves stuck the moment they have to apply the concepts on their own.
That's completely normal. Domain and range describe something specific about how functions work, and the idea needs a little time to settle before it starts to feel intuitive.
Our Mathnasium tutors put together this guide to explain what domain and range actually mean, how to find them from a graph and equation, and mistakes to watch out for along the way.
The domain of a function is the complete set of input values it can accept, every x-value that is allowed to go in. The range of a function is the complete set of output values it can produce, every y-value that can come out.
Think of a function as a vending machine. The domain is every valid button you can press. If a button is out of order, it gets excluded from the domain. The range is the complete set of snacks that can actually drop into the dispenser when you push those valid buttons. Some items might not be loaded.
Knowing which inputs are allowed and which outputs are possible is exactly what the domain and range are asking you to figure out.
Reading domain and range from a graph is the most visual approach, and for most students, it's the easiest place to start. We’ll show you how to do it, step by step.
Run your eyes across the graph horizontally, from the leftmost point to the rightmost point.
Every x-value the graph covers is part of the domain.
Ask yourself: where does the graph start and where does it end horizontally?
If it stretches in both directions with arrows, it has no left or right boundary, and the domain goes to infinity in both directions.
Now run your eyes vertically, from the lowest point the graph reaches to the highest. Every y-value the graph covers is part of the range.
Ask yourself:
What is the lowest point this graph hits?
What is the highest?
If the graph goes up or down without end, the range is unbounded in that direction.
Once you know what the domain and range cover, write them using interval notation.
Two rules to remember:
Square brackets [ ] mean the endpoint is included
Parentheses ( ) mean the endpoint is not included
Infinity always gets a parenthesis; you can never actually reach it, so it's never included
So if the domain covers all x-values from -3 onwards, including -3, you write [-3, ∞).
If the range goes up to but doesn't include 7, you write (-∞, 7).
If the graph has arrows at either end, the function continues beyond what's visible on the page.
Follow where the arrows point; if they go left, the domain extends to negative infinity. If they go right, it extends to positive infinity. The same logic applies vertically for range.
Keep in mind that if an arrow points diagonally (like down and to the left), it is moving toward negative infinity for both the domain and the range.
Before you write your answer, scan the graph for three things:
Holes: a single point shown as an open circle. The x-value at that point is excluded from the domain, and its corresponding y-value is excluded from the range.
Gaps: a section where the graph simply doesn't exist. Any x-values or y-values in that gap are excluded.
Asymptotes: Invisible (or dashed) boundary lines that the graph approaches but never touches. If a graph flattens out along a horizontal line or shoots vertically along a vertical line, those boundary values must be excluded using parentheses ( ).
Picture a parabola opening upward with its vertex sitting at the point (2, -4).
Running your eyes left to right, the graph stretches in both directions with no boundary; the domain is (-∞, ∞).
Running your eyes bottom to top, the lowest point is -4 at the vertex, and the graph goes upward from there without end; range is [-4, ∞).
Notice the square bracket at -4: the vertex is an actual point on the graph, so that value is included.
When there's no graph in front of you, the equation itself tells you everything you need to know.
When there's no graph in front of you, you find the domain and range algebraically, by analyzing the equation itself.
Let's walk through it together.
If the equation has a denominator, set it equal to zero and solve.
Whatever value of x makes the denominator zero is undefined; division by zero doesn't exist, so that value gets excluded from the domain.
For example, take f(x) = \(\Large\frac{1}{(x-3)}\)
Set the denominator equal to zero: x - 3 = 0, so x = 3. That means x = 3 is excluded.
The domain is every real number except 3, written as (-∞, 3) ∪ (3, ∞). The ∪ symbol just means "and also”; you're combining two separate intervals into one domain.
If the equation has a square root, the expression inside it cannot be negative; you can't take the square root of a negative number and get a real answer.
Set the expression inside the square root greater than or equal to zero and solve.
The values of x that satisfy that condition make up your domain.
For example, take f(x) = \(\sqrt{(x-5)}\).
Set up the inequality: x - 5 ≥ 0, so x ≥ 5.
That means only x-values of 5 and above are allowed. The domain is [5, ∞).
Square bracket at 5 because x = 5 is included, which makes the expression equal to zero, which is fine.
If the equation consists only of basic polynomials (like x2 or 2x + 1), there's nothing that can break it, there's nothing that can break it. Every x-value works as a valid input. The domain is (-∞, ∞)
The range is harder to read from an equation than the domain, because instead of asking what goes in, you're asking what can come out. The approach depends on the type of function you're working with.
Here are the ones you'll encounter most often in Algebra 2 and Pre-Calculus:
Linear functions like f(x) = 2x + 1 produce every possible y-value as x changes.
Range is (-∞, ∞).
Quadratic functions like f(x) = x² have a vertex that represents either a minimum or a maximum output. An upward parabola has a minimum; the range starts at the y-value of the vertex and goes up. A downward parabola has a maximum; the range goes down from the vertex. For f(x) = x², the vertex is at (0,0), so the range is [0, ∞).
Rational functions like f(x) = \(\Large\frac{1}{x}\) can produce any y-value except zero; the graph gets infinitely close to zero but never actually reaches it. Range is (-∞, 0) ∪ (0, ∞).
Square root functions like f(x) = \(\sqrt{x}\) can only produce outputs of zero or above, since square roots never give negative results. Range is [0, ∞).
Let’s use f(x) = \(\sqrt{\Large\frac{(x+2)}{(x-1)}}\) as an example, a function that combines both a square root and a fraction.
Work through it in order:
First, the square root: x + 2 ≥ 0, so x ≥ -2.
Then, the fraction: x - 1 ≠ 0, so x ≠ 1.
Put both restrictions together: x must be greater than or equal to -2, but x cannot equal 1. The domain is [-2, 1) ∪ (1, ∞).
That's the complete process.
For most algebraic functions, it comes down to two primary restrictions:
Is there a fraction?
Is there a square root?
Answer those, and you’ll find the domain.
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Before you finalize any domain or range answer, run through these checks.
Each one corresponds to a mistake that's easy to make and just as easy to avoid once you know to look for it.
Look carefully. Sometimes the fraction is nested inside a larger expression and is easy to miss.
If there is one, you haven't finished until you've set the denominator equal to zero and excluded that value.
A function can have both a fraction and a square root at the same time, like the example we used above.
Check for each one separately, then combine the restrictions. Students often catch one and miss the other.
Two things to verify: the smaller number always comes first, and infinity always gets a parenthesis.
A bracket next to an infinity symbol is one of the most common notation errors we see in Algebra 2, and it's an easy point to lose on a test for something that has nothing to do with whether you actually understood the math.
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