Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(g(x, y)=2 x^{2}-y^{2}\). Compute \(g(1,2), g(2,1), g(1,1)\), \(g(-1,1)\), and \(g(2,-1) .\)

Step-by-Step Solution

Answer

The values of the function at the given points are: \(g(1,2) = -2\), \(g(2,1) = 7\), \(g(1,1) = 1\), \(g(-1,1) = 1\), and \(g(2,-1) = 7\).

Step 1: 1. Plug the given (x, y) into the function

We will plug each of the five ordered pairs into the function \(g(x, y) = 2x^2 - y^2\).

Step 2: 2. Simplify

Now, let's simplify the function for each ordered pair, one by one: 1. For (1,2), we have \(g(1,2) = 2(1)^2 - (2)^2 = 2(1) - 4\). 2. For (2,1), we have \(g(2,1) = 2(2)^2 - (1)^2 = 2(4) - 1\). 3. For (1,1), we have \(g(1,1) = 2(1)^2 - (1)^2 = 2(1) - 1\). 4. For (-1,1), we have \(g(-1,1) = 2(-1)^2 - (1)^2 = 2(1) - 1\). 5. For (2,-1), we have \(g(2,-1) = 2(2)^2 - (-1)^2 = 2(4) - 1\).

Step 3: 3. Evaluate the function

Now, let's evaluate the function for each ordered pair: 1. For (1,2), we have \(g(1,2) = 2 - 4 = -2\). 2. For (2,1), we have \(g(2,1) = 8 - 1 = 7\). 3. For (1,1), we have \(g(1,1) = 2 - 1 = 1\). 4. For (-1,1), we have \(g(-1,1) = 2 - 1 = 1\). 5. For (2,-1), we have \(g(2,-1) = 8 - 1 = 7\). So, the values of the function at the given points are as follows: - \(g(1,2) = -2\) - \(g(2,1) = 7\) - \(g(1,1) = 1\) - \(g(-1,1) = 1\) - \(g(2,-1) = 7\)

Key Concepts

Algebra

Understanding variable manipulation and basic arithmetic operations like addition, subtraction, multiplication, and exponentiation is essential in algebra. When we evaluate functions such as the one given in the exercise, we need to replace variables with numbers and perform arithmetic. For instance, in the function \( g(x, y) = 2x^2 - y^2 \), the terms \( 2x^2 \) and \( y^2 \) illustrate algebraic manipulation with variables and constants.

• Plugging in Values: Substitute each ordered pair \( (x, y) \) into the function. In the given problem, we replace \( x \) and \( y \) with numeric values.
• Simplify Expressions: Follow the order of operations — exponents first, then multiplication and subtraction. This includes evaluating \( x^2 \), multiplying by 2, and subtracting \( y^2 \).

A strong grasp of algebra allows us to transform and evaluate expressions correctly, which is crucial for solving the given exercise accurately. By practicing these fundamental concepts, you can approach similar problems with confidence.

Coordinate Geometry

Coordinate geometry involves graphing and understanding geometric relationships using coordinates. This concept allows you to visualize algebraic equations graphically. Each ordered pair \((x, y)\) in this problem represents a point on the Cartesian plane.

• Ordered Pairs: Each solution pair like \((1,2)\) or \((2,-1)\) represents exact locations on the plane, showing how the function behaves at each point.
• Plotting Points: You can graph each of these points to see their distribution in the plane. This can help visualize how \( g(x, y) \) varies with different values of \( x \) and \( y \).

By understanding coordinate geometry, you can extend your analysis to observe the graphical behavior of functions across different regions of the plane. This visual approach not only aids intuition but also strengthens your grasp of algebraic solutions.

Mathematical Functions

Mathematical functions are rules that assign each input to exactly one output. In this exercise, \( g(x, y) = 2x^2 - y^2 \) is a function of two variables, \( x \) and \( y \). Functions are foundational in mathematics as they describe relationships between variables.

• Function Formulation: Understand that \( g(x,y) \) can represent many different scenarios based on real-world phenomena, providing outputs for given inputs as demonstrated in the exercise.
• Evaluation of Functions: Testing different inputs helps to understand the behavior of the function. For example, evaluating for \( (1,2) \) demonstrates the output when \( x = 1 \) and \( y = 2 \).

Mastering the concept of functions allows you to determine how changing variables affects outcomes, adding depth to your understanding of mathematical relationships and solving problems efficiently. It's a vital skill in diverse areas of mathematics and science.