Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Find the function values. \(f(x, y)=4-x^{2}-4 y^{2}\) (a) \(f(0,0)\) (b) \(f(0,1)\) (c) \(f(2,3)\) (d) \(f(1, y)\) (e) \(f(x, 0)\) (f) \(f(t, 1)\)

Step-by-Step Solution

Answer

The function values for the given inputs are (a) 4, (b) 0, (c) -36, (d) 3 - 4y^{2}, (e) 4 - x^{2}, (f) t^{2} - 4.

Step 1: Calculate Value for (a) \(f(0,0)\)

Substitute \(x=0\) and \(y=0\) into the function \(f(x, y)=4-x^{2}-4 y^{2}\). This gives us \(f(0,0) = 4-(0)^{2}-4*(0)^{2} = 4\).

Step 2: Calculate Value for (b) \(f(0,1)\)

Substitute \(x=0\) and \(y=1\) into the function. This gives us \(f(0,1) = 4-(0)^{2}-4*(1)^{2} = 4 - 4 = 0\).

Step 3: Calculate Value for (c) \(f(2,3)\)

Substitute \(x=2\) and \(y=3\) into the function. This gives us \(f(2,3) = 4-(2)^{2}-4*(3)^{2} = 4 - 4 - 36 = -36\).

Step 4: Calculate Value for (d) \(f(1,y)\)

Substitute \(x=1\) and leave \(y\) as a variable into the function. This results in \(f(1,y) = 4-(1)^{2}-4 y^{2} = 4 - 1 - 4y^{2} = 3 - 4y^{2}\).

Step 5: Calculate Value for (e) \(f(x,0)\)

Substitute \(y=0\) and leave \(x\) as a variable into the function. This results in \(f(x,0) = 4-x^{2}-4*(0)^{2} = 4 - x^{2}\).

Step 6: Calculate Value for (f) \(f(t,1)\)

Substitute \(x=t\) and \(y=1\) into function. This results in \(f(t,1) = 4-(t)^{2}-4*(1)^{2} = 4 - t^{2} - 4 = t^{2} - 4\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function when a particular set of variables are given. In multivariable calculus, this often involves substituting multiple variables into a function. Let's take a closer look at how this is done:

• First, identify the function you are working with. In this case, the function is given as \( f(x, y) = 4 - x^2 - 4y^2 \).
• Next, substitute the values for \( x \) and \( y \) into the function. For example, to find \( f(0,1) \), plug \( x=0 \) and \( y=1 \) into the function. You get \( f(0,1) = 4 - (0)^2 - 4(1)^2 \).
• Perform the algebraic operations: Calculate the squares and then subtract them from the constant, in this case, 4.

This process helps us determine specific outcomes of the function based on different inputs, providing insights into how the variables impact the function's behavior.

Two-Variable Functions

Two-variable functions involve equations where the output depends on two distinct variables. For example, in the function \( f(x, y) = 4 - x^2 - 4y^2 \), the values of both \( x \) and \( y \) affect the result.

• These functions help model complex situations in mathematics, where the outcome isn't reliant on a single variable.
• Graphically, a two-variable function can be represented in three dimensions, forming a surface in space.
• The changes in one variable, holding another constant, can create cross-sections of the function, offering clearer insights into its behavior.

Understanding two-variable functions is crucial in scenarios such as physics and economics, where multiple influencing factors must be considered simultaneously.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. When evaluating functions, manipulating algebraic expressions is a key step. Let's break down an example:

• Consider \( 4 - x^2 - 4y^2 \). This expression involves constants (like 4), variable terms (\( x^2 \) and \( 4y^2 \)), and the arithmetic operation of subtraction.
• The powers in the expression represent more complex forms of multiplication, where a variable is multiplied by itself.
• Simplifying these expressions involves following the order of operations: solving powers, then multiplication or division, and finally addition or subtraction.

Mastering algebraic expressions helps students approach more sophisticated problems by simplifying complex mathematical tasks into manageable steps.