Problem 15 - Calculus Early Transcendentals

Question

Find and sketch the domain of the function. $$f(x, y)=\sqrt{1-x^{2}}-\sqrt{1-y^{2}}$$

Step-by-Step Solution

Answer

The domain is the square region \((-1, -1)\) to \((1, 1)\) in the xy-plane.

Step 1: Analyze the Function's Formula

The function is given as \( f(x, y) = \sqrt{1-x^2} - \sqrt{1-y^2} \). This function involves square roots, which require non-negative arguments for real number outputs. We need both \( \sqrt{1-x^2} \geq 0 \) and \( \sqrt{1-y^2} \geq 0 \) to determine the domain.

Step 2: Set Conditions for Square Roots

For \( \sqrt{1-x^2} \) to be defined and real, the expression under the square root \( 1-x^2 \) must be greater than or equal to zero. The same applies to \( \sqrt{1-y^2} \). These requirements translate to \( 1-x^2 \geq 0 \) and \( 1-y^2 \geq 0 \).

Step 3: Solve Inequalities for x and y

The inequality \( 1-x^2 \geq 0 \) can be solved as \( -1 \leq x \leq 1 \), since subtracting from 1 gives \( x^2 \leq 1 \), leading to \( -1 \leq x \leq 1 \). Similarly, \( 1-y^2 \geq 0 \) leads to \( -1 \leq y \leq 1 \). So, the domain conditions are \( x, y \in [-1, 1] \).

Step 4: Understand the Domain Region

The solution implies that \( (x, y) \) must lie in the square region defined by \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\). This is the area where both \( \sqrt{1-x^2} \) and \( \sqrt{1-y^2} \) are defined within the real numbers.

Step 5: Sketch the Domain

To sketch the domain, draw a square in the coordinate plane where the vertices are at the points \((-1, -1), (-1, 1), (1, -1), (1, 1)\). The function is defined for every point within this square, including its border.

Key Concepts

Square Roots

Square roots are mathematical operations that find the original number that has been squared. Specifically, the square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For example, the square root of 9 is 3, because \( 3 \times 3 = 9 \). Square roots have specific requirements when dealing with real numbers:

• The number inside the square root (radicand) must be non-negative.
• If the radicand is negative, the square root is considered undefined in the real number system.

In the context of the function \( f(x, y) = \sqrt{1-x^2} - \sqrt{1-y^2} \), we need the expressions \( 1-x^2 \) and \( 1-y^2 \) to be at least zero for the square roots to be real. This ensures the function produces real number outputs.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where we can graphically display relationships between two variables. It is composed of two axes, the horizontal axis known as the x-axis and the vertical axis known as the y-axis.
Every point on this plane is identified by a pair of numerical coordinates \((x, y)\).
In this exercise, we use the coordinate plane to visualize the domain of the function \(f(x, y)\).By determining \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\),we can sketch the region where the function is defined. The domain forms a square in the coordinate plane, enabling us to see all points where the function output is real. This visualization helps in understanding constraints and ranges intuitively.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They can express situations where one value is greater, less, or has a potential range of values compared to another. Common symbols used include \(>\), \(<\), \(\geq\), and \(\leq\).
For the function \(f(x, y) = \sqrt{1-x^2} - \sqrt{1-y^2}\),we use inequalities to determine the conditions under which the square roots are valid, \(1-x^2 \geq 0\) and \(1-y^2 \geq 0\).When solved, these inequalities give the relationships \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\),restricting \(x\) and \(y\) to these intervals. These inequalities provide the necessary conditions for the domain, guiding us to understand which points in the plane keep the function defined.

Real Numbers

Real numbers include all the numbers we encounter in everyday life, such as integers, fractions, and irrational numbers. They can be positive, negative, or zero, and are represented on the number line. Real numbers have no imaginary parts.
For functions, ensuring outputs are real numbers is critical since many operations rely on them.
In our exercise, the goal is to identify the domain where the function \(f(x, y) = \sqrt{1-x^2} - \sqrt{1-y^2}\)is defined and yields real numbers as results. By satisfying the conditions \(1-x^2 \geq 0\) and \(1-y^2 \geq 0\),we guarantee these output values are real. This makes the concept of real numbers fundamental when seeking to understand the limits within which such functions operate.