Question

Group Activity In Exercises \(37-40\) , verify that the function is continuous and state its domain. Indicate which theorems you are using, and which functions you are assuming to be continuous. $$y=\frac{1}{\sqrt{x+2}}$$

Short answer

The function \(y=\frac{1}{\sqrt{x+2}}\) is continuous and its domain is \([-2, ∞)\)

Step-by-step solution

Define the function

First, we consider the function \(y=\frac{1}{\sqrt{x+2}}\). This is a composite function, which is a combination of two functions, the reciprocal function and the square root function.

Consider the Domains of the Inner and Outer Functions

Next, we consider the domain of each of the two functions individually. The reciprocal function is continuous for all real numbers except zero, and the square root function is continuous for all non-negative numbers (meaning zero and positive numbers). The composite function is continuous when both of its individual functions are continuous.

Determine the Domain of the Composite Function

By considering the domains of the reciprocal and square root functions that are combined into the composite function, we find that the composite function is continuous when the square root function is non-negative (so \(x+2 ≥ 0\)) and when the reciprocal function is not zero (so \(x+2 ≠ 0\)). This means that x must be greater than or equal to -2. Therefore, the domain of the function is \([-2,∞)\). Since the function is defined and continuous at every point in its domain, we can say the function is continuous everywhere in its domain.

Applying the Composition of Continuous Functions Theorem

We are making use of the Composition of Continuous Functions Theorem, which states that the composition of two continuous functions is also continuous. If \(f\) and \(g\) are two functions such that \(f(g(x))\) is defined, then \(f(g(x))\) is continuous at \(x\) if \(g\) is continuous at \(x\) and \(f\) is continuous at \(g(x)\)