Problem 26 - 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. $$f(-3,0)$$

Step-by-Step Solution

Answer

\(-6\)

Step 1: Substitute the given values of x and y into the function

We are asked to compute the value of the function \(f(-3,0)\). So we will substitute \(x=-3\) and \(y=0\) into the function: $$ f(-3,0)=2(-3)+3(0)-6(-3)(0) $$

Step 2: Perform arithmetic operations

Now, perform the arithmetic operations in the expression: $$ f(-3,0)=(-6)+0-0 $$

Step 3: Simplify the expression

Lastly, simplify the expression to get the final result: $$ f(-3,0)=-6 $$ Thus, the value of the function \(f(-3,0)\) is \(\boxed{-6}\).

Key Concepts

Function Evaluation

Understanding function evaluation is crucial when working with mathematical functions. A function can be thought of as a machine that takes some inputs and provides a unique output based on a specific rule or formula. In this case, we have a function defined as \( f(x, y) = 2x + 3y - 6xy \). Evaluating a function involves substituting the values of the variables \(x\) and \(y\) into the function's formula to calculate the output. For example, to compute \(f(-3,0)\), you substitute \(-3\) for \(x\) and \(0\) for \(y\) into the expression, resulting in:\[ \begin{align*} f(-3,0) & = 2(-3) + 3(0) - 6(-3)(0) \end{align*} \] This substitution is the first step in determining the value of the function for specific inputs.

Arithmetic Operations

Arithmetic operations are the basic mathematical processes used to compute numerical expressions. These include addition, subtraction, multiplication, and division. In the context of our function \(f(x, y) = 2x + 3y - 6xy\), after substituting the values for \(x\) and \(y\), you must perform arithmetic operations to simplify the expression to one single numerical value. For instance, when evaluating \(f(-3,0)\), the substitution gives us \(2(-3)\), \(3(0)\), and \(-6(-3)(0)\).

• Perform \(2(-3)\) which equals \(-6\).
• Calculate \(3(0)\) which results in \(0\).
• Lastly, \(-6(-3)(0)\) simplifies to \(0\).

The calculated values are then combined according to the arithmetic operations specified: \[ \begin{align*} f(-3,0) & = (-6) + 0 - 0 \t &= -6 \end{align*} \] Thus, you see the expression simplifies through a series of arithmetic operations, resulting in the final computed value of \(-6\).

Integer Functions

An integer function is a type of function where the domain and codomain are integers, meaning it takes integer inputs and provides integer outputs. In the context of this exercise, the function \(f : \mathbf{Z} \times \mathbf{Z} \rightarrow \mathbf{Z}\) is designed to accept any integers \(x\) and \(y\) and output an integer result. The expression \(f(x, y) = 2x + 3y - 6xy\) ensures that no matter which integers you plug in for \(x\) and \(y\), the operations of multiplication and addition preserve the property of producing an integer result. Calculating \(f(-3,0)\) with straightforward arithmetic operations adheres to this principle, resulting in an integer output, \(-6\), reflecting the consistency and reliability of integer functions in discrete mathematics.