Problem 27 - Discrete Mathematics with Applications

Question

Let \(f: \mathbf{Z} \times \mathbf{Z} \rightarrow \mathbf{Z}\) defined by \(f(x, y)=2 x+3 y-6 x y .\) Compute the following. $$f(-2,3)$$

Step-by-Step Solution

Answer

\(f(-2, 3) = 41\)

Step 1: Substitute values

First, we will substitute the values of x and y into the function \(f(x, y)\):$$ f(-2, 3) = 2(-2) + 3(3) - 6(-2)(3) $$ Step 2: Perform Arithmetic

Step 2: Arithmetic Calculations

Now, we will perform the arithmetic calculations:$$ f(-2, 3) = -4 + 9 + 36 $$ Step 3: Simplify the Expression

Step 3: Simplification

Finally, we will simplify the expression to get the value of the function:$$ f(-2, 3) = -4 + 9 + 36 = 41 $$ So, $$ f(-2,3) = 41 $$.

Key Concepts

Integer Functions

Integer functions are a type of function where both the input and output values are integers. In the given exercise, the function \( f(x, y) = 2x + 3y - 6xy \) takes two integer inputs \( x \) and \( y \) and returns an integer output.
These functions are extensively used in discrete mathematics and computer science because they naturally handle integer data types, which are common in algorithms and programming. Integer functions can have various forms, including linear, polynomial, or more complex expressions.
Working with integer functions often involves arithmetic operations, where we apply the function rules using integer arithmetic. The resulting output is predictable, which makes integer functions ideal for mathematical modeling and computational processes. Understanding integer functions is fundamental to solving problems involving integer sequences, optimizations, and discrete structures.

Arithmetic Operations

Arithmetic operations are basic mathematical calculations such as addition, subtraction, multiplication, and division. These operations are crucial for evaluating expressions, especially in integer functions.
In the provided exercise, the function \( f(x, y) = 2x + 3y - 6xy \) incorporates several arithmetic operations:

• Multiplication: The expressions \( 2x \), \( 3y \), and \( 6xy \) involve multiplying numbers.
• Addition and Subtraction: The terms are combined using addition and subtraction to form the complete function expression.

When substituting specific values for \( x \) and \( y \), these arithmetic operations are applied systematically to compute the result.
Performing these calculations requires attention to detail to ensure the right application of each operation and maintain the correct order, often following the order of operations (parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right)) to get accurate results.

Function Evaluation

Function evaluation is the process of finding the output of a function for specific input values. This involves substituting the input values into the function's formula and performing necessary arithmetic operations to simplify and solve the expression.
In the example \( f(x, y) = 2x + 3y - 6xy \), we are asked to evaluate \( f(-2, 3) \). Here, we replace \( x \) with \(-2\) and \( y \) with \( 3 \), which gives us \( 2(-2) + 3(3) - 6(-2)(3) \).
Next, we carry out the arithmetic calculations:

• Multiply: \( 2(-2) = -4 \), \( 3(3) = 9 \), and \(- 6(-2)(3) = 36 \).
• Add and subtract: combine these values to get \(-4 + 9 + 36 \), which equals \( 41 \).

This process demonstrates how function evaluation directly connects inputs to outputs through specific mathematical operations. Successful function evaluation delivers clear insights into how the function behaves and provides concrete solutions to mathematical exercises.