HesapHanem

Domain and Range Calculator

Pick a function type and enter its coefficient and constant.

How it is calculated

Choose one of four function families and the tool reports the domain (allowed x) and range (possible y) by rule, not by graphing. Linear y = ax + c: domain all real numbers, range all real numbers, or just y = c if a = 0. Quadratic y = ax² + c: domain all real numbers; the vertex sits at y = c, so the range is y ≥ c when a > 0, y ≤ c when a < 0, and y = c when a = 0. Square root y = √x + c: domain x ≥ 0, range y ≥ c; a is not used. Reciprocal y = a/x + c: domain x ≠ 0, range y ≠ c, or y = c if a = 0. A blank a defaults to 1 and a blank c to 0. Any other type gives no result. The quadratic has no bx term.

Worked example

Select quadratic with a = −2 and c = 3, the function y = −2x² + 3. Nothing restricts x, so the domain is all real numbers. Since x² is never negative, −2x² is never positive, so the highest y is reached at x = 0, giving y = 3. Because a is negative the parabola opens downward and the range is y ≤ 3. The vertex (0, 3) is the maximum, and y falls without limit as x moves away from zero.

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Common mistakes

  • Expecting a bx term. The quadratic here is only ax² + c, so for x² + 4x + 1 the tool cannot give the true vertex or range; complete the square first.
  • Mixing up c with the vertex's x-coordinate. In this model c is the vertical shift, so the range boundary is y = c, not x = c.
  • Reading y ≥ c for a downward parabola. The sign of a decides: positive a gives y ≥ c, negative a gives y ≤ c.
  • Entering a slope for the square-root type and expecting it to matter. The tool treats it as √x + c, so a is ignored.
  • Applying the square-root result to a shifted function such as √(x − 2). A horizontal shift changes the domain in a way the tool does not show.

About this calculator

The domain of a function is the complete set of input values (x) for which it is defined, and the range is the complete set of output values (y) it can produce — together they describe exactly what a function can accept and what it can return, which is foundational before graphing, composing, or inverting a function.

Each common function type has a characteristic domain and range shape: a linear function (y = mx + b, m ≠ 0) has all real numbers for both; a quadratic (here y = ax² + c) has all reals as domain but a range bounded above or below by its vertex at y = c, depending on the sign of a; a square root function (here y = √x + c) is restricted to inputs that keep the radicand non-negative, so its domain is x ≥ 0 and its range is y ≥ c; and a reciprocal function (here y = a/x + c) excludes x = 0 from the domain, since division by zero is impossible, and excludes y = c from the range (y = 0 when c = 0). This calculator identifies the domain and range for a chosen function type from its coefficient a and constant c; the square root type does not use a, and square roots opening in other directions are not covered.

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Comments (1)

  • Editorial team

    The constant c only moves the graph up or down, and the square-root type uses just the form root of x plus c. For c = 2 it reports x ≥ 0 and y ≥ 2, but a shifted root such as the root of x minus 3 would need x ≥ 3, which this tool does not model. Slide a horizontal shift into the domain by hand before trusting the answer. Which function families should we add next?

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