Problem 69 - Calculus and its applications

Question

Is the function given by \(G(x)=\frac{1}{x-1}\) continuous over the interval \((0, \infty) ?\) Why or why not?

Step-by-Step Solution

Answer

The function \(G(x)=\frac{1}{x-1}\) is not continuous over \((0, \infty)\) because it is undefined at \(x = 1\).

Step 1: Understand the Function

The function given is \(G(x) = \frac{1}{x-1}\). This is a rational function, meaning it's the ratio of two polynomials. Rational functions are generally continuous wherever they are defined, except at points where the denominator is zero, which could create a division by zero (undefined points).

Step 2: Find the Undefined Points

The denominator of \(G(x)\) is \(x - 1\). To find the points where the function is undefined, set the denominator equal to zero: \(x - 1 = 0\). Solving this equation, we find that the function is undefined at \(x = 1\) because the denominator is zero at this point, causing a division by zero.

Step 3: Analyze the Interval \((0, \infty)\)

The interval in question is \((0, \infty)\), which includes all positive numbers greater than zero. The number 1 is within this interval; thus, \(x = 1\) is a point where the function is not defined—hence not continuous—within the given interval.

Step 4: Conclusion about Continuity

A function is considered continuous over an interval if it is continuous at every single point in that interval. Since \(G(x)\) is not defined at \(x = 1\), it is not continuous over \((0, \infty)\). The point \(x = 1\) is a discontinuity in this interval.

Key Concepts

Rational Functions

Rational functions are a fundamental concept in calculus and algebra. A rational function is typically expressed as the ratio of two polynomials, such as \(G(x) = \frac{1}{x-1}\). The numerator can be any polynomial, and the denominator must also be a polynomial that is not zero. Since we have division involved, it's crucial these functions are looked at carefully,

• Unlike simple polynomials, rational functions can have points where they are undefined.
• These are usually where the denominator equals zero, causing a break or gap in the graph, known as discontinuities.
• Such points make these functions not continuous across their domain.

Rational functions have many applications in real-world scenarios where relationships exhibit varying rates of change. Understanding where these functions break helps us anticipate and interpret their behavior.

Interval Notation

Interval notation is a method used to represent a range of values on the number line. This concise method is essential for indicating where functions are defined or continuous.
For example, consider the interval

• \((0, \infty)\): This includes all positive real numbers greater than zero up to infinity.

In this notation:

• Parentheses \(()\) signify that the endpoint is not included in the interval, meaning every number in the interval is greater than but not equal to the endpoint.
• Brackets \([]\) would be used to include the endpoint. For instance, \([1, \infty)\) includes 1 but extends to infinity.

Understanding interval notation helps to easily grasp where a function is defined and how continuity plays out within certain regions.

Division by Zero

Division by zero is a key mathematical idea, particularly significant in understanding rational functions. When you encounter a division by zero:

• The operation is undefined because you cannot divide a quantity into zero parts.
• It creates points of discontinuity in rational functions, where the function cannot be calculated.

In the case of the function \(G(x) = \frac{1}{x-1}\), when \(x = 1\), the denominator becomes zero, leading to an undefined value. This consequently impacts the function's continuity, causing a break or hole in the graph at that point.
By identifying and understanding these undefined points, you can better predict and explain the behavior of rational functions, ensuring the analysis of domain and range is accurate.