Problem 12 - Calculus Early Transcendentals

Question

Find and sketch the domain of the function. $$f(x, y)=\sqrt{x y}$$

Step-by-Step Solution

Answer

The domain consists of the first and third quadrants, including the x-axis and y-axis.

Step 1: Identify the Constraint for Square Roots

The function is given as \(f(x, y) = \sqrt{x y}\). Since this involves a square root, we must ensure that the expression inside the square root is non-negative to make the function real-valued. Thus, the condition for the domain is \(x y \geq 0\).

Step 2: Determine the Condition for Products

The inequality \(x y \geq 0\) implies that both \(x\) and \(y\) must be of the same sign or at least one of them can be zero. This divides the domain into the following regions: 1. \(x > 0\) and \(y > 0\)2. \(x < 0\) and \(y < 0\)3. \(x = 0\) 4. \(y = 0\)

Step 3: Visualize the Domain on the Coordinate Plane

Plot the different regions on a two-dimensional coordinate plane. The domain includes the first and third quadrants, as well as the x-axis and y-axis:- First Quadrant: Both \(x > 0\) and \(y > 0\) (including the origin).- Third Quadrant: Both \(x < 0\) and \(y < 0\).- x-axis: Where \(y = 0\).- y-axis: Where \(x = 0\).

Step 4: Sketch the Domain

Draw the coordinate plane and shade the areas representing each condition:- Shade the first quadrant (where both \(x\) and \(y\) are positive).- Similarly, shade the third quadrant (where both \(x\) and \(y\) are negative).- The entire x-axis and y-axis should be highlighted as they are part of the domain.

Key Concepts

Domain of a Function

The domain of a multivariable function refers to the set of all possible input values that allow the function to produce real-valued outputs.
For instance, in the function given by \(f(x, y) = \sqrt{x y}\), the domain is determined by the constraints of the square root operation.
In mathematics, square roots are defined only for non-negative numbers to keep the results real. Thus, the expression inside the square root, \(xy\), must be greater than or equal to zero.

• \(x y \geq 0\) means the product of \(x\) and \(y\) cannot be negative.
• This breaks down into situations where either both \(x\) and \(y\) are positive or negative.
• Additionally, if either \(x\) or \(y\) is zero, the product is zero, which is allowed.

Understanding the domain helps us visualize where the function is "defined" or "valid" on a coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graphically represent mathematical functions, including those with multiple variables.
This plane consists of two perpendicular lines, called the x-axis and y-axis, which intersect at the origin (0,0).
The coordinate plane allows for easy visualization of relationships between variables and is particularly useful for identifying the domain of multivariable functions.Points on the plane are expressed as ordered pairs \((x, y)\), representing their position relative to the axes:

• The x-axis is horizontal, running left to right (or vice versa).
• The y-axis is vertical, running top to bottom (or vice versa).
• Quadrants are the four "sections" into which the plane is divided by the axes.

Visualizing functions like \(f(x, y) = \sqrt{x y}\) requires recognizing which quadrants or lines (axes) satisfy the domain conditions. By shading specific areas, we can indicate where the function is defined.

Square Roots

Square roots are operations that take a number and return its original value when multiplied by itself. They can only handle non-negative inputs in real number systems.
If we denote the expression inside a square root as \(z\), then \(\sqrt{z}\) is defined if \(z \geq 0\).
This is because negative numbers inside a square root lead to imaginary numbers, which are not real.For the function \(f(x, y) = \sqrt{x y}\), understanding square roots involves:

• Recognizing that \(xy \geq 0\) ensures you can perform the operation.
• The square root of zero is zero, simplifying some parts of the domain analysis.

Real-valued square roots simplify calculations and ensure that functions remain within the realm of real numbers. This is crucial for properly sketching and analyzing domains of multivariable functions.

Inequalities in Functions

Inequalities in functions are used to determine the range of inputs that meet specific conditions.
For the function \(f(x, y) = \sqrt{x y}\), the inequality \(x y \geq 0\) is crucial for establishing its domain.
This inequality can be further broken down into conditions that tell us about the signs of \(x\) and \(y\).The inequality implies:

• Both \(x\) and \(y\) can be positive, leading us to the first quadrant of the coordinate plane.
• Both \(x\) and \(y\) can be negative, placing us in the third quadrant.
• Either variable can be zero, encompassing the x-axis and y-axis.

Understanding inequalities within the context of functions helps in mapping out the regions where the functions are valid. This framework allows us to trace and highlight the domain over the coordinate plane, ensuring accurate depiction and problem-solving.