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The Domain and Range Calculator analyzes six fundamental function types and identifies their domain restrictions, range boundaries, asymptotes, and critical points. Understanding the domain and range of a function is one of the most important skills in mathematics — the domain is the set of all valid input values, and the range is the set of all possible output values. Every function has domain and range constraints that dictate where the function exists and what values it can produce.
For a linear function $$f(x) = ax + b$$, both the domain and range span all real numbers ($$-\infty, \infty$$) — there are no restrictions on input or output. For a quadratic function $$f(x) = ax^2 + bx + c$$, the domain is still all reals, but the range is restricted by the vertex: if $$a > 0$$ the parabola opens upward and the range is $$[y_{\text{vertex}}, \infty)$$; if $$a < 0$$ it opens downward and the range is $$(-\infty, y_{\text{vertex}}]$$.
The square root function $$f(x) = a\sqrt{x - h} + k$$ requires the radicand to be non-negative, so the domain is $$[h, \infty)$$. Its range depends on the sign of $$a$$: if $$a > 0$$, the range is $$[k, \infty)$$; if $$a < 0$$, it is $$(-\infty, k]$$. The rational function $$f(x) = \frac{a}{x - h} + k$$ is undefined at $$x = h$$ (vertical asymptote) and approaches but never reaches $$y = k$$ (horizontal asymptote), so its domain is $$(-\infty, h) \cup (h, \infty)$$ and its range is $$(-\infty, k) \cup (k, \infty)$$.
The logarithmic function $$f(x) = a\ln(x - h) + k$$ requires a positive argument, giving domain $$(h, \infty)$$ with a vertical asymptote at $$x = h$$, while its range is all reals. Conversely, the exponential function $$f(x) = ae^{x-h} + k$$ has domain all reals but a horizontal asymptote at $$y = k$$ that bounds the range.
These six function families cover the vast majority of functions encountered in algebra, precalculus, and introductory calculus. Knowing their domain and range patterns is essential for graphing, solving equations, composing functions, finding inverses, and applying functions to real-world modeling. This calculator computes the key domain and range boundaries, identifies asymptotes, locates the vertex or critical point, and evaluates the function at a test point so you can verify that your test value lies within the valid domain.
Whether you are checking homework, preparing for an exam, or exploring how transformations (shifts and stretches) affect domain and range, this tool provides instant, clear results. The critical point output is particularly useful for quadratic functions (vertex), square root functions (endpoint), and rational/logarithmic functions (asymptote locations).
Select the function type and enter the coefficients. The calculator analyzes the function's algebraic structure to determine:
For linear and exponential functions with unrestricted domains, the "Domain starts at" field shows 0 as a neutral indicator — the actual domain is all reals. For square root and logarithmic functions, this value is the left endpoint of the domain; inputs below this value are undefined. The range boundary indicates the minimum (for upward-opening parabolas and positive square roots) or maximum (for downward-opening parabolas) output value. Asymptotes are values the function approaches infinitely closely but never equals. If f(test point) shows 0 unexpectedly, your test point may be outside the domain.
Inputs
Results
Domain: [3, ∞) because x − 3 ≥ 0. Range: [1, ∞) because the minimum output is k = 1 (when x = 3). f(7) = 2√4 + 1 = 5.
Inputs
Results
Domain: (−∞, 2) ∪ (2, ∞) — undefined at x = 2. Range: (−∞, 5) ∪ (5, ∞) — horizontal asymptote at y = 5. f(4) = 3/2 + 5 = 6.5.
The domain is the set of all valid input values ($$x$$-values) for which the function produces a real output. Common restrictions include: division by zero (excluded), negative values under even roots (excluded), and non-positive values inside logarithms (excluded).
The range is the set of all possible output values ($$y$$-values) the function can produce. For example, $$f(x) = x^2$$ has range $$[0, \infty)$$ because a square is never negative. The range depends on both the function type and its specific coefficients.
You can square any real number, so the domain is unrestricted. However, a parabola has a minimum (opens up) or maximum (opens down) at its vertex, which bounds the range. For $$f(x) = x^2$$, the minimum output is 0 at the vertex, so the range is $$[0, \infty)$$.
An asymptote is a line that the function approaches but never touches or crosses (in most cases). A vertical asymptote at $$x = h$$ means the function heads toward $$\pm\infty$$ as $$x \to h$$. A horizontal asymptote at $$y = k$$ means the function approaches $$k$$ as $$x \to \pm\infty$$.
Horizontal shifts (replacing $$x$$ with $$x - h$$) shift the domain by $$h$$ units. Vertical shifts (adding $$k$$) shift the range by $$k$$ units. Vertical stretches (multiplying by $$a$$) scale the range, and if $$a < 0$$, the range direction reverses. These transformations apply uniformly across all function types.
A 0 in the asymptote fields for functions without asymptotes (like linear or quadratic) means that field is not applicable — those functions have no asymptotes. For test point evaluation, a 0 may indicate the point is outside the domain (e.g., testing x = 1 for $$\sqrt{x - 3}$$). Check that your test point satisfies the domain conditions.
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