Problem 20 - Calculus Early Transcendentals

Question

Find the domain of the following functions. $$h(x, y)=\sqrt{x-2 y+4}.$$

Step-by-Step Solution

Answer

Answer: The domain of the function \(h(x,y)=\sqrt{x-2y+4}\) is the set of all \((x,y)\) coordinates where \(y \leq \frac{1}{2}x - 2\). In other words, the domain is all points \((x,y)\) that lie below the line \(y=\frac{1}{2}x-2\).

Step 1: Analyze the inequality we need to solve to find the domain

To find the domain of the function \(h(x,y)=\sqrt{x-2y+4},\) we need to ensure that the expression inside the square root is non-negative. So we need to solve the inequality \(x-2y+4 \geq 0.\)

Step 2: Rearrange the inequality to solve for \(y\)

We want to solve the inequality for \(y\) so that we can find the domain in terms of \((x,y)\) coordinates. To do this, we can rearrange the inequality as follows: 1. Subtract \(x\) from both sides: \(-2y + 4 \geq -x\) 2. Divide both sides by \(-2\) and reverse the inequality sign: \(y \leq \frac{1}{2}x - 2\)

Step 3: Determine the domain of the function

Based on our rearranged inequality, the domain of the function \(h(x,y)=\sqrt{x-2y+4}\) is the set of all \((x,y)\) coordinates where \(y \leq \frac{1}{2}x - 2\). In other words, the domain is all points \((x,y)\) that lie below the line \(y=\frac{1}{2}x-2\).

Key Concepts

Understanding Inequalities in Domains

Inequalities are essential when determining the domain of functions, especially those involving expressions under a square root. The reason is simple: you cannot take the square root of a negative number in the real number system. So, for a function like \(h(x, y) = \sqrt{x - 2y + 4}\), the expression \(x - 2y + 4\) must be greater than or equal to zero.
This leads us to form an inequality: \(x - 2y + 4 \geq 0\).
Solving this inequality helps us identify where the function is defined.

• Start by rearranging the terms to isolate \(y\).
• Perform basic algebraic steps, such as subtraction and division.
• Always remember to reverse the inequality sign when dividing by a negative number.

The solution provides us with the conditions under which the function is valid, essentially describing a region in the plane where the function can "live."

Exploring Multivariable Functions

Multivariable functions, like \(h(x, y)\), involve more than one input. These functions can map points from a two-dimensional space (a plane) to real numbers. Understanding multivariable functions requires considering how changes in one variable affect the output when another is held constant.
For example, in the function \(h(x, y) = \sqrt{x - 2y + 4}\), both \(x\) and \(y\) influence the result simultaneously.
To better understand these functions, you should:

• Visualize the function output as a surface rather than a curve.
• Identify how constraints, such as inequalities, confine the domain to specific regions.
• Consider how these 2D conditions change as either \(x\) or \(y\) is altered.

Multivariable functions often require you to think geometrically, as solutions describe areas or spaces rather than simple lines or intervals.

The Role of the Square Root Function

The square root function is unique because it is only defined for non-negative inputs. When dealing with a multivariable function like \(h(x, y) = \sqrt{x - 2y + 4}\), it ensures that the expression inside the root stays zero or positive.
This requirement is why identifying the correct domain through inequalities is crucial.
If \(x - 2y + 4\) falls below zero, the square root becomes undefined, limiting the possible inputs.
Working with square roots requires careful balancing between solving algebraic inequalities and understanding the function's geometric implications.

• Remember: The square root function is restrictive and limits domain possibilities.
• Even a slight alteration in the expression inside can drastically change the valid \((x,y)\) points.
• Visual representation helps illustrate where the function can exist on a plane.

The square root serves as a gatekeeper, shaping the domain by deciding which values are permitted.