calculator domain and range

What are domain and range?

In algebra, every function has two important sets: the domain and the range. The domain is all valid input values (x-values) that the function can accept. The range is all possible output values (y-values) the function can produce. Understanding both helps you graph correctly, solve equations faster, and avoid impossible inputs.

Domain in plain language

Ask: “Which x-values are allowed?” For example, in a square root function, you cannot take the square root of a negative number (in real numbers), so inputs may be restricted. In rational functions, you can’t divide by zero, so certain x-values are excluded.

Range in plain language

Ask: “What y-values can come out?” A quadratic opening up has a lowest y-value (its vertex), while an exponential function shifted upward can never cross its horizontal asymptote. Those facts define the range.

How to use this calculator

• Select a function family from the dropdown.
• Enter the parameters shown (like a, h, k or m, b).
• Click Calculate Domain & Range.
• Read the interval notation and any restrictions provided.

Domain and range rules by function type

1) Linear: f(x) = mx + b

• Domain: all real numbers, always.
• Range: all real numbers if m ≠ 0; a single value {b} if m = 0.

2) Quadratic: f(x) = a(x − h)² + k

• Domain: all real numbers.
• Range: y ≥ k if a > 0, y ≤ k if a < 0, and {k} if a = 0.

3) Absolute value: f(x) = a|x − h| + k

• Domain: all real numbers.
• Range: y ≥ k if a > 0, y ≤ k if a < 0, and {k} if a = 0.

4) Square root: f(x) = a√(x − h) + k

• Domain: x ≥ h.
• Range: y ≥ k if a > 0, y ≤ k if a < 0, and {k} if a = 0.

5) Reciprocal: f(x) = a/(x − h) + k

• Domain: all real numbers except x = h.
• Range: all real numbers except y = k (if a ≠ 0).

6) Exponential: f(x) = a·b^(x − h) + k

• Domain: all real numbers.
• Range: y > k if a > 0, y < k if a < 0, and {k} if a = 0.

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

• Domain: x > h.
• Range: all real numbers (if a ≠ 0), otherwise {k}.

Interval notation quick reference

• (−∞, ∞): all real numbers
• [c, ∞): all values greater than or equal to c
• (−∞, c]: all values less than or equal to c
• (−∞, h) ∪ (h, ∞): all real numbers except h
• {k}: exactly one value k

Common mistakes students make

• Forgetting to exclude values that make denominators zero.
• Confusing horizontal shift h with vertical shift k.
• Using parentheses when brackets are needed in interval notation.
• Assuming every function has range (−∞, ∞).

Final takeaway

Domain and range are not just textbook vocabulary—they describe what a function is allowed to do. If you learn the transformation forms and restrictions for each family, you can determine domain and range quickly without graphing every single time. Use the calculator above to check your work and build intuition.