Problem 5 - Calculus with Concepts in Calculus

Question

Find the domain of the function. \(g(x, y)=\sqrt{x^{2}+y^{2}-25}\)

Step-by-Step Solution

Answer

The domain of the function is outside or on the circle of radius 5 centered at the origin.

Step 1: Identify the Domain Constraint

The given function is a square root function, which is defined only for non-negative values inside the square root. Thus, we need the expression inside the square root, \(x^2 + y^2 - 25\), to be greater than or equal to zero. This gives us the inequality: \x^2 + y^2 - 25 \geq 0.\

Step 2: Rearrange the Inequality

Rearranging the inequality \(x^2 + y^2 - 25 \geq 0\), we get: \x^2 + y^2 \geq 25.\

Step 3: Interpret the Inequality

The inequality \(x^2 + y^2 \geq 25\) represents the region outside or on the boundary of a circle with a radius of 5, centered at the origin \((0, 0)\). Thus, the domain of the function is all points \(x, y\) such that \x^2 + y^2 \geq 25.\

Key Concepts

Square Root Function

The square root function is a common mathematical operation. It involves finding a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 multiplied by 3 equals 9. In function form, with variables, it is more complex. Consider a function like \(g(x, y) = \sqrt{x^2 + y^2 - 25}\). Here, the expression inside the square root, \(x^2 + y^2 - 25\), must be non-negative (greater than or equal to zero) for the square root to be defined.
This is because the square root of a negative number is not a real number, making it undefined within real number arithmetic, which we typically work with. Therefore, the arguments of the square root function must be handled carefully. These conditions, determining the allowable inputs, lead directly into discussions of domain, ensuring the function's definition across all its possible values.

Inequality Constraints

An inequality constraint is a restriction that specifies the set of values a variable can satisfy. Deriving from mathematical inequalities, it limits the region under which a formula can be applied. For the function at hand, \(x^2 + y^2 - 25 \geq 0\), becomes the central constraint. By rearranging, we see \(x^2 + y^2 \geq 25\).
This requires all solutions to lie within specified limits. Inequalities are used to determine regions in space that are "valid" or permissible. Here, the inequality indicates that the sum of the squares of \(x\) and \(y\) must meet or exceed 25. This mathematical condition forms the boundary of a geometric shape which is crucial for defining the domain of functions in a spatial sense.

Geometry of Circles

Understanding the geometry of circles is fundamental when interpreting equations like \(x^2 + y^2 = r^2\). This equation describes a circle with a center at the origin \(0, 0\) and a radius \(r\). In our case, it sets the groundwork for understanding the function's domain.
For \(x^2 + y^2 \geq 25\), we are dealing with a circle of radius 5 centered at \(0, 0\). The inequality implies we consider all points outside this circle, as well as those on its boundary.

• The boundary condition \(x^2 + y^2 = 25\) defines the circle itself.
• The inequality portion \(x^2 + y^2 > 25\) adds all points beyond the circle.

This scenario is visually represented by a filled region extending outward from the circle, encapsulating the domain of the function. Recognizing these geometric interpretations allows us to apply restrictions and understand the spatial configuration of solutions to the function.