Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(f(x, y)=x^{2}+2 x y-x+3\). Compute \(f(1,2), f(2,1)\), \(f(-1,2)\), and \(f(2,-1)\).

Step-by-Step Solution

Answer

The values of the function for the provided pairs of values are as follows: \(f(1, 2) = 7\), \(f(2, 1) = 9\), \(f(-1, 2) = 1\), and \(f(2, -1) = 1\).

Step 1: Write down the function expression

First, we need to properly understand the mathematical function that we're given. It is expressed as: \[f(x, y) = x^{2} + 2xy - x + 3\]

Step 2: Calculate \(f(1, 2)\)

Now we will substitute the values \(x = 1\) and \(y = 2\) into the function and perform the calculations: \[f(1, 2) = (1)^{2} + 2(1)(2) - (1) + 3 = 1 + 4 - 1 + 3 = 7\]

Step 3: Calculate \(f(2, 1)\)

Substitute the values \(x = 2\) and \(y = 1\) into the function and perform the calculations: \[f(2, 1) = (2)^{2} + 2(2)(1) - (2) + 3 = 4 + 4 - 2 + 3 = 9\]

Step 4: Calculate \(f(-1, 2)\)

Substitute the values \(x = -1\) and \(y = 2\) into the function and perform the calculations: \[f(-1, 2) = (-1)^{2} + 2(-1)(2) - (-1) + 3 = 1 - 4 + 1 + 3 = 1\]

Step 5: Calculate \(f(2, -1)\)

Substitute the values \(x = 2\) and \(y = -1\) into the function and perform the calculations: \[f(2, -1) = (2)^{2} + 2(2)(-1) - (2) + 3 = 4 - 4 - 2 + 3 = 1\] The values of the function for the provided pairs of values are as follows: \[f(1, 2) = 7, f(2, 1) = 9, f(-1, 2) = 1, f(2, -1) = 1\]

Key Concepts

Polynomial functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables, multiplied by coefficients. They are fundamental in algebra and appear frequently in science and engineering.
A polynomial function can have just one variable, like \(f(x) = x^2 + 3x + 2\), or multiple variables, as in our case: \(f(x, y) = x^{2} + 2xy - x + 3\).
Key features of polynomial functions include:

• Coefficients: Numbers in front of the variables, such as the 2 in 2xy.
• Exponents: Powers like the 2 in \(x^2\), which must be whole numbers.
• Degree: The highest degree of its terms (for example, \(x^2\) is degree 2).

These functions are closely connected with many mathematical techniques and provide foundational concepts for calculus and beyond.

Function evaluation

Evaluating a function essentially means finding its output for specific input values. With polynomial functions of two variables like \(f(x, y) = x^{2} + 2xy - x + 3\), the process involves substituting the given values for \(x\) and \(y\) and simplifying.
For example, when evaluating \(f(1, 2)\):

• Substitute \(x = 1\) and \(y = 2\) into the function.
• Perform each operation step by step.
• Simplify it to reach \(7\).

Repeating these steps for each pair of \((x, y)\) given in the problem allows us to understand how the function behaves at different points. This step-by-step substitution is crucial for accuracy and understanding the output fully.

Mathematical calculations

Mathematical calculations involve executing operations like addition, subtraction, multiplication, and applying exponents. When dealing with polynomial functions such as \(f(x, y) = x^{2} + 2xy - x + 3\), it's important to follow the order of operations:
1. Perform calculations inside any parentheses first.
2. Handle exponents and powers next.
3. Execute multiplication and division from left to right.
4. Finally, do addition and subtraction from left to right.
For instance, when evaluating \(f(-1, 2)\), follow each operation with care by:

• Calculating the exponent: \((-1)^2 = 1\)
• Multiplying: \(2(-1)(2) = -4\)
• Addressing each component to ensure all operations are completed accurately.

This meticulous approach to calculation ensures that we avoid errors and achieve the correct answer, demonstrating the critical importance of mastering basic mathematical operations.