Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(f(x, y)=2 x+3 y-4\). Compute \(f(0,0), f(1,0), f(0,1)\), \(f(1,2)\), and \(f(2,-1) .\)

Step-by-Step Solution

Answer

The computed function values for the given points are: \(f(0, 0) = -4\), \(f(1, 0) = -2\), \(f(0, 1) = -1\), \(f(1, 2) = 4\), and \(f(2, -1) = -3\).

Step 1: Compute f(0, 0)

To compute the function value at the point (0, 0), we replace x and y with 0 in the function: \(f(0, 0) = 2(0) + 3(0) - 4\) Now, we can calculate the result: \(f(0, 0) = -4\)

Step 2: Compute f(1, 0)

To compute the function value at the point (1, 0), we replace x with 1 and y with 0 in the function: \(f(1, 0) = 2(1) + 3(0) - 4\) Now, we can calculate the result: \(f(1, 0) = -2\)

Step 3: Compute f(0, 1)

To compute the function value at the point (0, 1), we replace x with 0 and y with 1 in the function: \(f(0, 1) = 2(0) + 3(1) - 4\) Now, we can calculate the result: \(f(0, 1) = -1\)

Step 4: Compute f(1, 2)

To compute the function value at the point (1, 2), we replace x with 1 and y with 2 in the function: \(f(1, 2) = 2(1) + 3(2) - 4\) Now, we can calculate the result: \(f(1, 2) = 4\)

Step 5: Compute f(2, -1)

To compute the function value at the point (2, -1), we replace x with 2 and y with -1 in the function: \(f(2, -1) = 2(2) + 3(-1) - 4\) Now, we can calculate the result: \(f(2, -1) = -3\) The computed function values are as follows: - \(f(0, 0) = -4\) - \(f(1, 0) = -2\) - \(f(0, 1) = -1\) - \(f(1, 2) = 4\) - \(f(2, -1) = -3\)

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics, where you determine the output of a function given specific inputs. It involves replacing the variables in an expression with given numbers and simplifying the result.

Consider the function from our exercise, expressed as \(f(x, y) = 2x + 3y - 4\). When you evaluate this function at a particular pair of \(x\) and \(y\) values, you substitute these values into the function's equation and calculate the result.

For instance, to find \(f(0, 0)\), you would replace \(x\) with 0 and \(y\) with 0, resulting in \(f(0, 0) = 2(0) + 3(0) - 4 = -4\). This process is applied to any pair \((x, y)\) to obtain the corresponding output from the function. Mastering function evaluation allows you to understand the behavior of the function across different input values, which is essential in graphing functions and solving equations.

Linear Functions

Linear functions are algebraic expressions where each term is either a constant or the product of a constant and a single variable. These functions produce a straight line when graphed on a coordinate plane, and they have the general form \(f(x) = mx + b\) for a single variable, or \(f(x, y) = ax + by + c\) for two variables, where \(a\), \(b\), and \(c\) are constants.

In the case of our exercise, the function \(f(x, y) = 2x + 3y - 4\) is a linear function of two variables, \(x\) and \(y\). It represents a plane in three-dimensional space. To understand a linear function, it's helpful to know that the coefficients of \(x\) and \(y\) (2 and 3 in this case) affect the slope of the line in their respective dimensions, and the constant term (-4 here) represents the y-intercept when \(x\) and \(y\) are both zero.

Linear functions are simple yet powerful tools in applied mathematics, economics, physics, and engineering, as they can model relationships with a constant rate of change.

Applied Mathematics

Applied mathematics involves utilizing mathematical methods and concepts to solve real-world problems across various fields such as science, engineering, business, and technology. It differs from pure mathematics, which focuses more on theoretical aspects and abstract concepts without immediate practical application.

Our exercise is an example of applied mathematics, as evaluating the function at different points may correspond to analyzing real situations. For instance, in a business context, \(f(x, y)\) could represent the profit from selling \(x\) units of product A and \(y\) units of product B. Computing \(f(0, 0)\), \(f(1, 0)\), \(f(0, 1)\), etc., would yield the profit outcomes for different sales scenarios.

Understanding the concept of evaluating functions and linear functions is crucial since these are the building blocks for more complex mathematical modeling in applied mathematics, allowing for analysis, prediction, and decision-making based on mathematical data.