Problem 72 - Calculus and its applications

Question

Is the function given by \(F(x)=\frac{3}{x-5}\) continuous on \(\mathbb{R} ?\)

Step-by-Step Solution

Answer

The function is not continuous on \( \mathbb{R} \); it is discontinuous at \( x = 5 \).

Step 1: Identify the Form of the Function

We start by examining the function given as \[ F(x) = \frac{3}{x-5} \]This function is a rational function, which is generally continuous everywhere it is defined.

Step 2: Determine Points of Discontinuity

A rational function \( \frac{f(x)}{g(x)} \) is discontinuous whenever the denominator is zero. Set the denominator equal to zero to find the points of discontinuity:\[ x-5 = 0 \]Solving for \( x \), we have:\[ x = 5 \]Thus, \( F(x) \) is discontinuous at \( x = 5 \).

Step 3: Examine the Domain of the Function

The given function \( F(x) = \frac{3}{x-5} \) is undefined at \( x = 5 \) due to division by zero. This means it cannot be part of its domain. The function is defined for all other real numbers. Therefore, the domain is:\[ \mathbb{R} \setminus \{5\} \]

Step 4: Conclusion on Continuity

A function is continuous at a point if it is defined at that point, and its limit exists and equals the function's value at that point. Since \( F(x) \) is undefined at \( x = 5 \), it is not continuous everywhere on \( \mathbb{R} \). Thus, \( F(x) \) is not continuous on all of \( \mathbb{R} \).

Key Concepts

Rational Function

A rational function is a type of function that is the ratio of two polynomial functions. Generally, it is expressed in the form \( R(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial functions. The function \( F(x) = \frac{3}{x-5} \) is a perfect example of a rational function, consisting of a constant numerator and a simple linear expression in the denominator.
Rational functions are known for being continuous at all points where they are defined. However, they are not defined where the denominator is zero, because division by zero is undefined. Thus, while rational functions offer a wide scope of continuity over their domain, identifying the discontinuities caused by the denominator is crucial for understanding their complete behavior.
Key characteristics to remember about rational functions include:

• Their general form as a ratio of polynomials.
• Continuity across their domain, except at points where the denominator equals zero.
• The potential for asymptotic behavior near points of discontinuity.

Points of Discontinuity

In the context of rational functions, points of discontinuity are locations in the function's domain where the function ceases to be continuous. Such points generally arise where the denominator of the rational function equals zero. For the function \( F(x) = \frac{3}{x-5} \), the denominator \( x-5 \) becomes zero when \( x = 5 \).
At \( x = 5 \), the function encounters a point of discontinuity because the value of the function is undefined due to division by zero this causes an interruption in the function's smooth plot. Discontinuities in rational functions often lead to vertical asymptotes, where the function approaches but never actually reaches a particular value.
When dealing with rational functions, make sure to:

• Set the denominator equal to zero to discover points of discontinuity.
• Be aware of vertical asymptotes that occur at these points.
• Understand that these points are outside the function's domain.

Domain of a Function

The domain of a function refers to the set of all possible input values \( x \) that allow the function to produce a real output. For rational functions like \( F(x) = \frac{3}{x-5} \), it is crucial to determine which values of \( x \) result in a defined function? Typically, these are all real numbers except where the denominator is zero.
For \( F(x) = \frac{3}{x-5} \), the function is undefined where \( x = 5 \), due to division by zero. Thus, the domain of this particular function excludes \( x = 5 \). It can be expressed in set notation as \( \mathbb{R} \setminus \{5\} \), meaning all real numbers except 5.
When investigating the domain of a function, keep in mind:

• Identify values that lead to undefined expressions, typically involving division by zero.
• Consider the behavior and limits of the function around points of discontinuity.
• The precise domain dictates where the function is, and isn't, able to be used.