Problem 3 - College Algebra with Modeling and Visualization

Question

Evaluate the expression for the given \(f(x, y)\). $$ f(2,-3) \text { if } f(x, y)=x^{2}+y^{2} $$

Step-by-Step Solution

Answer

The value of \( f(2, -3) \) is 13.

Step 1: Substitute the given values into the function

We are given the function \( f(x, y) = x^2 + y^2 \) and need to find \( f(2, -3) \). Substitute \( x = 2 \) and \( y = -3 \) into the function to evaluate it.

Step 2: Square the individual terms

Calculate \( 2^2 \) and \( (-3)^2 \). \( 2^2 = 4 \) and \( (-3)^2 = 9 \).

Step 3: Add the squared terms

Add the results from the previous step: \( 4 + 9 = 13 \). This is the value of \( f(2, -3) \).

Key Concepts

Evaluate Expression

Evaluating an expression means calculating its value for given variables. In our exercise, we have the function \( f(x, y) = x^2 + y^2 \), and we need to find the result for the specific case \( f(2, -3) \). To evaluate this, we substitute the given values into the expression. The goal is to simplify the expression completely so that we can obtain a final numerical result. Evaluating involves basic arithmetic operations like squaring numbers and then adding them. This process helps translate abstract expressions into concrete values, making them easier to understand and use.

Function Notation

Function notation is a way to describe mathematical functions in a compact and clear manner. In the expression \( f(x, y) \), \( f \) represents the function, while \( x \) and \( y \) are its variables. This notation tells us that \( f \) is a rule or formula that relates the input values (\( x \) and \( y \)) to an output result. The expression \( x^2 + y^2 \) specifies the exact calculations to perform with the inputs.

• The letter \( f \) can be replaced by any letter, not just 'f'. It's commonly used because it stands for 'function'.
• The variables \( x \) and \( y \) are called independent variables. They're what you plug into the function.
• The output from the function, after performing the calculations, is called the dependent variable, depending on the values of \( x \) and \( y \).

The use of function notation facilitates complex mathematical expressions and makes it easy to communicate and work with these relationships.

Substitution

Substitution is a method used to simplify expressions and solve problems by replacing variables with given or known values. In our task, we substitute \( x = 2 \) and \( y = -3 \) into the function \( f(x, y) = x^2 + y^2 \). This process involves replacing every instance of the variable \( x \) with 2, and every instance of \( y \) with -3.
Substitution helps convert a general problem into a specific one by using certain values to make calculations possible. Here's a simple substitution process:

• Identify which values need to be substituted into which variables.
• Replace each variable in the expression with its corresponding value.
• Proceed with arithmetic calculations to simplify the expression.

Through substitution, complex functions and algebraic expressions are broken down into simpler arithmetic steps, making solutions achievable and understandable.