Question

Give a precondition and a postcondition for the predefined function sqrt, which returns the square root of its argument.

Short answer

Answer: The precondition for the sqrt function is that the input argument must be a non-negative number (x ≥ 0). The postcondition is that the output (result) is always a non-negative number, and it is the correct square root of the input argument (y ≥ 0 and y² = x).

Step-by-step solution

Precondition for sqrt

The precondition for the square root function is that the input argument must be a non-negative number (i.e., a positive number or zero). In other words, the input number should not be a negative number. This is essential because the square root of a negative number is undefined in the real number system. Mathematically, the precondition can be represented as: \[ x \geq 0 \]

Postcondition for sqrt

The postcondition for the square root function is that the output (result) is always a non-negative number, and it is the correct square root of the input argument. This means that when the output is squared, it should be equal to the input argument. Mathematically, the postcondition can be represented as: \[ \sqrt{x} = y \Rightarrow y \geq 0 \text{ and } y^2 = x \]

Key concepts

Mathematical Functions

Mathematical functions are rules that assign each input to exactly one output. They are essential for building relationships between variables and solving real-world problems.
In mathematics, a function is typically represented as \(f(x)\), where \(x\) is the input, and \(f(x)\) is the output.
Functions can model various scenarios, such as population growth over time or the trajectory of a projectile.

Key properties of functions include:

• Domain: The set of all possible input values for which the function is defined.
• Range: The set of all possible output values the function can produce.
• Uniqueness: Each input should only map to a single output.

Understanding how functions work helps in analyzing and predicting outcomes based on different inputs. With mathematical functions, we apply logical rules to achieve computational efficiency and precision.

Square Root Function

The square root function is a special kind of mathematical function. It represents the value \(y\) such that when you square \(y\), you get the input number back.
For example, the square root of 9 is 3 because \(3^2 = 9\).
The function is denoted as \(\sqrt{x}\). A unique feature of the square root function is its relationship with non-negative numbers. Square roots of numbers are always non-negative because a negative \(y\) would result in a positive number when squared, which contradicts the absolute value output.

Properties of the square root function:

• Non-negativity: For any non-negative input, the square root will also be non-negative.
• Inverse Function: The square root is the inverse of squaring a number.
• Non-linearity: The square root function does not have a constant rate of change.

It is used in numerous fields, including geometry, physics, and finance, helping in calculating distances, energy, and financial risk assessments.

Non-negative Numbers

Non-negative numbers include all positive numbers and zero. They are written as \(x \geq 0\).
These numbers form a crucial part of mathematics due to their properties and uses in computations and real-world applications.
Non-negative numbers ensure certainty in results of many functions, such as the square root function, that require specific input conditions.

Characteristics of non-negative numbers:

• Always Greater or Equal to Zero: They can never be negative.
• Includes Whole Numbers: All whole numbers are non-negative.
• Supports Real-World Applications: Non-negative numbers are used in contexts where negative quantities don't make sense, such as measuring height or weight.

Recognizing non-negative numbers' role ensures proper input for specific functions and avoids errors during computations. They are foundational in mathematical operations and everyday problem-solving.