Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{x^{2}-1}{x}\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x^{2} - 1}{x}\) is continuous on the intervals (-∞,0) and (0,∞). It is discontinuous at \(x = 0\) because the function is not defined at this point.

Step 1: Identify the Undefined Point

The function is undefined when the denominator, \(x\), is zero. So the function is undefined at \(x = 0\).

Step 2: Simplify the Function

We can simplify the function \(f(x) = \frac{x^{2} - 1}{x}\) to \(f(x) = x - \frac{1}{x}\) since \(\frac{x^{2}}{x} = x\).

Step 3: Determine the Continuity of the Simplified Function

The function \(f(x) = x - \frac{1}{x}\) is continuous for all real numbers except \(x = 0\). The limit exists at every point where the function is defined. Also, the function value is equal to the limit value everywhere it is defined.

Step 4: Identify the Discontinuous Point

Based on the analysis from step 1, we know that the function is discontinuous at \(x = 0\). Therefore, at this point, the conditions of continuity are not satisfied because the function is not defined at \(x = 0\).

Key Concepts

Understanding Limits

Limits are a fundamental aspect of calculus and continuity of functions. Imagine a limit as the value that a function approaches as the input, or independent variable, gets closer to a specified point. It's like predicting where a function is heading even if it doesn't exactly land there.
In the context of the function given, which is originally expressed as \[ f(x) = \frac{x^2 - 1}{x} \], you try to determine what the value of the function will be when approaching certain values of \( x \).
In cases where the function is undefined, like at \( x = 0 \) here, you can still discuss the limit of \( f(x) \) as \( x \) approaches a value, even if the function doesn't exactly reach a point. When a function is unpredictable or sharply changes at a point, it may not have a limit there, indicating potential discontinuity.
To check continuity, observe if the function approaches the same value from both sides of a point. If it does, we say it has a limit at that point—and potentially continuity!

Exploring Function Discontinuity

An important aspect of continuity involves identifying points of discontinuity in a function. These are specific points where a function does not behave nicely or predictably. Discontinuities can occur for different reasons like division by zero, which is evident in our problem when \( x = 0 \).
For a function to be continuous at a point \( c \), three conditions must be satisfied:

• The function \( f(x) \) must be defined at \( x = c \).
• The limit of the function \( f(x) \) as \( x \) approaches \( c \) must exist.
• The limit of \( f(x) \) as \( x \) approaches \( c \) must equal \( f(c) \).

In the given function, \[ f(x) = \frac{x^2 - 1}{x}, \]discontinuity happens at \( x = 0 \) because the function is not defined there, meaning the above conditions do not hold. This undefined behavior results in discontinuity, and hence \( f(x) \) needs re-evaluation or consideration of its simplified form away from \( x = 0 \).

Navigating Rational Functions

Rational functions are quotients of two polynomials, just like our given function \[ f(x) = \frac{x^2 - 1}{x}, \] where the numerator is \( x^2 - 1 \) and the denominator is \( x \).
These functions can often be continuous wherever they are defined, but points of discontinuity can emerge at values that make the denominator zero, which must never happen as division by zero is undefined. Tendencies to simplify these expressions, as seen where our example turns into \[ f(x) = x - \frac{1}{x} \] (for \( x eq 0 \)), often reveals underlying characteristics and gives insights into behavior and continuity.
When simplifying a rational function, it helps to identify points where they might not be behaving predictably, like \( x = 0 \) in our case. These insights allow one to consider the broader implications of what the graph of the function might look like, aiding in deeper understanding.