Problem 52 - Calculus and its applications

Question

Is the function given by \(F(x)=\sqrt{x}\) continuous at \(x=-1 ?\) Why or why not?

Step-by-Step Solution

Answer

The function \( F(x) = \sqrt{x} \) is not continuous at \( x = -1 \) because it is not defined there.

Step 1: Understanding Function Requirements

To determine the continuity of the function at a given point, we must first check that the function is defined at that point. The function given is \( F(x) = \sqrt{x} \). Consider whether this function is defined at \( x = -1 \).

Step 2: Function Definition Check

For \( F(x) = \sqrt{x} \), the function is only defined for \( x \geq 0 \) since the square root of a negative number is not a real number. Thus, \( F(x) \) is not defined at \( x = -1 \).

Step 3: Conclusion of Continuity Analysis

A function is continuous at a point only if it is defined at that point, and the limit as \( x \) approaches the point equals the function value at that point. Since \( F(x) \) is not defined at \( x = -1 \), it cannot be continuous there.

Key Concepts

Square Root Function

The square root function, represented as \( F(x) = \sqrt{x} \), is a mathematical function that involves finding the principal square root of \( x \). The operation essentially asks, "What number multiplied by itself gives \( x \)?"

• This function typically deals with non-negative numbers.
• The domain where the square root function produces real numbers is \( x \geq 0 \).

This is because the square root of a negative number is not a real number in the context of standard real-number arithmetic. Rather, it is considered an imaginary number. When you graph \( F(x) = \sqrt{x} \), it starts from the origin (0,0) and forms a curve that moves upward to the right. This curve only resides in the first quadrant of a graph. Understanding this limitation is crucial when determining the function's behavior and continuity on real numbers.

Function Definition

Function definition involves specifying the mathematical rule or expression that a function uses to produce outputs from inputs. A vital aspect of any function is the domain, which consists of all input values for which the function is defined.In mathematics, a function is a relationship between sets that assigns each element from the first set (domain) to exactly one member of the second set (range). For \( F(x) = \sqrt{x} \):

• The domain is \( x \geq 0 \), represented by all non-negative real numbers.
• The range is \( y \geq 0 \), as the square root of any real number is non-negative.

If a function is to be continuous at a point, it must be defined at that point. This was crucial in determining whether \( F(x) = \sqrt{x} \) was continuous at \( x = -1 \). Since the function is not defined for negative values, \( F(x) \) cannot be continuous there.

Limits in Calculus

Limits in calculus are used to describe the behavior of a function as the input approaches a specific value. Limits are fundamental because they form the basis of defining continuity, derivatives, and integrals.For a function to be continuous at a point:

• The function must be defined at that point.
• The limit of the function as the input approaches the point must exist.
• The limit must equal the function's value at that point.

If any of these conditions are not met, the function is not continuous at the particular point. In the case of \( F(x) = \sqrt{x} \) at \( x = -1 \), the function wasn't defined there due to domain restrictions. This means that the first condition of continuity is not satisfied, preventing any further consideration of limits to establish continuity.