Domain Range Function Calculator

Domain and Range Function Calculator

Use this calculator to determine the domain and range for common types of functions based on their parameters. Select a function type and input the required coefficients.

Select Function Type: Rational: f(x) = 1 / (Ax + B) Square Root: f(x) = sqrt(Ax + B) Quadratic: f(x) = Ax² + Bx + C

Coefficient A:

Coefficient B:

Coefficient C:

Results:

Domain:

Range:

Understanding Domain and Range

In mathematics, the domain of a function refers to the set of all possible input values (often represented by x) for which the function is defined and produces a real output. The range of a function is the set of all possible output values (often represented by y or f(x)) that the function can produce from its domain.

Why are Domain and Range Important?

Function Definition: They precisely define where a function exists and what values it can yield.
Graphing: Understanding domain and range helps in accurately sketching the graph of a function.
Real-World Applications: In physics, engineering, and economics, domain and range often represent physical constraints or meaningful output values (e.g., time cannot be negative, distance cannot be negative).
Avoiding Undefined Operations: They help identify values that would lead to mathematical impossibilities, such as division by zero or taking the square root of a negative number.

Common Restrictions on Domain:

While polynomial functions generally have a domain of all real numbers, other function types have specific restrictions:

Rational Functions (Fractions): The denominator cannot be equal to zero. For example, in f(x) = 1 / (x - 3), x ≠ 3.
Square Root Functions (Even Roots): The expression under an even root (like a square root) must be greater than or equal to zero. For example, in f(x) = sqrt(x + 2), x + 2 ≥ 0, so x ≥ -2.
Logarithmic Functions: The argument of a logarithm must be strictly greater than zero. For example, in f(x) = log(x - 1), x - 1 > 0, so x > 1.

Determining Range:

The range is often determined by analyzing the behavior of the function, its graph, or by considering the domain and any transformations. For instance:

For f(x) = x², the domain is (-∞, ∞), but since squaring any real number results in a non-negative value, the range is [0, ∞).
For f(x) = sqrt(x), the domain is [0, ∞), and the range is also [0, ∞).
For f(x) = sin(x), the domain is (-∞, ∞), but its values oscillate between -1 and 1, so the range is [-1, 1].

Examples Using the Calculator:

Let's walk through some examples using the calculator above:

Example 1: Rational Function

Function Type: Rational: f(x) = 1 / (Ax + B)
Input A: 2
Input B: -4
Calculation: The denominator 2x - 4 cannot be zero. So, 2x - 4 ≠ 0, which means 2x ≠ 4, and x ≠ 2. The range for 1/something (where something is not zero) will never be zero.
Result:
- Domain: (-∞, 2) U (2, ∞)
- Range: (-∞, 0) U (0, ∞)

Example 2: Square Root Function

Function Type: Square Root: f(x) = sqrt(Ax + B)
Input A: -3
Input B: 9
Calculation: The expression under the square root, -3x + 9, must be greater than or equal to zero. So, -3x + 9 ≥ 0. Subtracting 9 from both sides gives -3x ≥ -9. Dividing by -3 and reversing the inequality sign gives x ≤ 3. The square root of a non-negative number always yields a non-negative result.
Result:
- Domain: (-∞, 3]
- Range: [0, ∞)

Example 3: Quadratic Function

Function Type: Quadratic: f(x) = Ax² + Bx + C
Input A: 1
Input B: -6
Input C: 5
Calculation: For any quadratic function, the domain is always all real numbers. To find the range, we need the y-coordinate of the vertex. The vertex formula for y is C - B² / (4A). Here, 5 - (-6)² / (4 * 1) = 5 - 36 / 4 = 5 - 9 = -4. Since A is positive (1 > 0), the parabola opens upwards, meaning the minimum value is at the vertex.
Result:
- Domain: (-∞, ∞)
- Range: [-4, ∞)