range domain calculator

What Is Domain and Range?

In algebra, the domain is the set of all valid x-values you can plug into a function. The range is the set of all output y-values the function can produce. Understanding both tells you where a function exists and what values it can reach.

If you are solving homework problems, checking graph behavior, or building intuition for precalculus and calculus, domain and range are essential first steps.

How to Use This Calculator

• Select a function family (linear, quadratic, reciprocal, square root, etc.).
• Enter transformation parameters a, h, and k.
• For exponential and logarithmic functions, enter base b.
• Click Calculate Domain & Range to see interval notation and a short interpretation.

This tool uses standard transformation forms often taught in Algebra 2 and Precalculus, making it easy to connect formulas to graph behavior.

Domain and Range Rules by Function Family

Linear: f(x) = a(x - h) + k

Domain is all real numbers. Range is all real numbers unless a = 0, in which case the function is constant and range is just {k}.

Quadratic: f(x) = a(x - h)² + k

Domain is all real numbers. The vertex is at (h, k). If a > 0, the parabola opens upward and range is [k, ∞). If a < 0, it opens downward and range is (-∞, k].

Absolute Value: f(x) = a|x - h| + k

Domain is all real numbers. Like quadratics, the minimum or maximum occurs at y = k depending on the sign of a.

Reciprocal: f(x) = a/(x - h) + k

x = h is excluded from the domain (vertical asymptote). y = k is excluded from the range (horizontal asymptote) when a ≠ 0.

Square Root: f(x) = a√(x - h) + k

Inside the square root must be nonnegative, so x ≥ h. Range depends on a: if a > 0 then y ≥ k; if a < 0 then y ≤ k.

Exponential: f(x) = a·b^{(x - h)} + k

Domain is all real numbers. If a > 0 then outputs are above k; if a < 0 then outputs are below k. Base must satisfy b > 0 and b ≠ 1.

Logarithmic: f(x) = a·log_b(x - h) + k

Input to the log must be positive, so x > h. For nonzero a, range is all real numbers. Base again requires b > 0 and b ≠ 1.

Common Mistakes to Avoid

• Forgetting to exclude undefined x-values (like division by zero).
• Ignoring root restrictions (square roots require nonnegative radicands).
• Using an invalid base for logs/exponentials (base 1 or negative bases).
• Confusing domain and range interval notation endpoints.

Why This Matters

Domain and range help with graphing, solving equations, checking model validity, and interpreting real-world constraints. In applications (physics, economics, engineering), knowing valid input/output limits often matters more than the equation itself.