Problem 18 - Calculus Early Transcendentals

Question

\(15-20\) Explain why the function is discontinuous at the given number a. Sketch the graph of the function. \(f(x)=\left\\{\begin{array}{ll}{\frac{x^{2}-x}{x^{2}-1}} & {\text { if } x \neq 1} \\ {1} & {\text { if } x=1}\end{array}\right. \quad a=1\)

Step-by-Step Solution

Answer

The function is discontinuous at \( x = 1 \) because the limit \( \frac{1}{2} \neq f(1) = 1 \).

Step 1: Check Function Definition

The function is defined as piecewise: - \( f(x) = \frac{x^2 - x}{x^2 - 1} \) for \( x eq 1 \).- \( f(x) = 1 \) for \( x = 1 \).We need to examine the behavior of the function near \( x = 1 \).

Step 2: Simplify Expression for x ≠ 1

The expression \( \frac{x^2 - x}{x^2 - 1} \) can be simplified by factoring:- Numerator: \( x^2 - x = x(x-1) \).- Denominator: \( x^2 - 1 = (x+1)(x-1) \).Thus, \( \frac{x^2 - x}{x^2 - 1} = \frac{x(x-1)}{(x+1)(x-1)} \).Cancel \( (x-1) \) from numerator and denominator (for \( x eq 1 \)):\[ \frac{x}{x+1} \] for \( x eq 1 \).

Step 3: Check Limit as x Approaches 1

Assess the limit of \( \frac{x}{x+1} \) as \( x \) approaches 1:\[ \lim_{{x \to 1}} \frac{x}{x+1} = \frac{1}{1+1} = \frac{1}{2} \].

Step 4: Compare Limit to Function Value at x = 1

The limit approaching \( x = 1 \) from both sides is \( \frac{1}{2} \), but the defined value of the function at \( x = 1 \) is \( f(1) = 1 \).Because the limit \( \frac{1}{2} eq 1 \), the function is discontinuous at \( x = 1 \).

Step 5: Sketch the Graph of the Function

The sketch highlights:- The function follows \( f(x) = \frac{x}{x+1} \) for \( x eq 1 \), a curve approaching \( y = \frac{1}{2} \) as \( x \to 1 \).- A removable discontinuity (hole) at \( x = 1 \) due to cancellation of \( (x-1) \).- A separate point plotted at \( x = 1, y = 1 \), indicating the defined function value not matching the limit.

Key Concepts

Piecewise Functions

A piecewise function is like a story with different chapters, where each chapter follows its own plot. In mathematics, this means the function is defined by different expressions over different intervals of its domain.
For example, the function in this context has two pieces:

• For all values of \( x \) except 1, the function is defined as \( f(x) = \frac{x^2 - x}{x^2 - 1} \).
• At \( x = 1 \), it is explicitly defined as \( f(x) = 1 \).

This is useful when a function is expected to model scenarios that change dramatically over its domain. Piecewise definitions ensure precision in capturing sudden changes or defining exceptions, like at \( x = 1 \) in this exercise.

Limits and Continuity

Limits and continuity are like the secret behind knowing what happens as we approach a wall without actually hitting it. In mathematics, this is crucial to understand what's happening to the function as it gets close to a certain point.
For our piecewise function, we look at the limit of \( \frac{x}{x+1} \) as \( x \) approaches 1:

• The limit calculation gave us \( \frac{1}{2} \), showing where \( f(x) \) "wants" to be when nearing 1.
• However, because \( f(1) = 1 \), an abrupt change happens at \( x = 1 \), causing a break in the graph's continuity.

Thus, for a function to be continuous at a point, the limit as we approach that point must equal the function's actual value at it.

Removable Discontinuity

A removable discontinuity is like a hiccup in the flow of function. It appears when a function could be continuous, but isn't due to a hole in the graph.
In our scenario, the original expression \( \frac{x^2 - x}{x^2 - 1} \) simplifies to \( \frac{x}{x+1} \) when \( x eq 1 \).
This simplification process involves canceling out \( (x-1) \) from top and bottom, effectively creating a removable hole at \( x = 1 \).

• The removable discontinuity doesn't allow the function to seamlessly connect at \( x = 1 \), because the limit \( \frac{1}{2} \) doesn't match the defined value \( f(1) = 1 \).

Properly addressing these holes means redefining the function at such points, ensuring continuity through the graph.

Graphing Functions

Graphing functions can resemble creating visual stories. Each element of the graph provides a glimpse into the behavior of the function across its domain. To graph our piecewise function, several steps must be taken:

• Draw the curve \( \frac{x}{x+1} \), noting it gets closer to \( y = \frac{1}{2} \) as \( x \) approaches 1.
• Identify the hole at \( x = 1 \), showing the break in continuity.
• Position a separate point at \( (1, 1) \), highlighting where the function is explicitly defined at \( x = 1 \).

Through graphing, students can visually understand where a function continues smoothly and where disruptions (like holes) occur, deepening their comprehension of functional behavior.