Functions
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Domain & Range

Which x-values go in, and which y-values come out?

The domain of a function is the set of all valid inputs (x-values). The range is the set of all possible outputs (y-values). For f(x) = √x, the domain is x ≥ 0 because you cannot take the square root of a negative number.

On a graph, the domain is the horizontal extent of the curve, and the range is its vertical extent. Some functions have restricted domains — holes, asymptotes, or square roots that cut off part of the graph.

Ask the AI anything — try "What is the domain of 1/(x − 2)?" or "Why can't the range of x² include negative numbers?"

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FAQ

What is domain?
The domain is the set of all x-values for which the function is defined. For example, √x has domain x ≥ 0 (no negative inputs), and 1/x has domain all real numbers except x = 0 (division by zero is undefined).
What is range?
The range is the set of all y-values the function can produce. For x², the range is y ≥ 0 because a square is never negative. For √x, the range is also y ≥ 0.
How do I find domain from a graph?
Look at how far left and right the curve extends along the x-axis. If the curve stops at a certain x-value, that is a domain boundary. Vertical asymptotes (dashed lines) mark x-values excluded from the domain.
What does "all real numbers except" mean?
It means the function works for every x-value except specific ones where it breaks — like division by zero or taking the log of zero. We write this as x ∈ ℝ, x ≠ 2 or (−∞, 2) ∪ (2, ∞).

Related Topics

What is a Function?Rational FunctionsAbsolute Value Functions