Problem 19 - Calculus: Early Transcendentals

Question

Explain why the function is discontinuous at the given number \( a \). Sketch the graph of the function. \( f(x) = \left\\{ \begin{array}{ll} x + 3 & \mbox{if \) x \le -1 \(} \hspace{40mm} a = -1\\\ 2^x & \mbox{if \) x > -1 \(} \end{array} \right.\)

Step-by-Step Solution

Answer

The function is discontinuous at \( x = -1 \) because the left-hand and right-hand limits differ at this point.

Step 1: Understand the Function

The function \( f(x) \) is a piecewise function with two parts: \( x + 3 \) for \( x \le -1 \) and \( 2^x \) for \( x > -1 \). The point of interest is \( a = -1 \), where the piece of the function changes.

Step 2: Evaluate the Left-Hand Limit

Determine the limit of \( f(x) \) as \( x \) approaches -1 from the left. Since \( f(x) = x + 3 \) for \( x \le -1 \), substitute \( x \) with -1: \[ \lim_{{x \to -1^-}} f(x) = -1 + 3 = 2. \]

Step 3: Evaluate the Right-Hand Limit

Determine the limit of \( f(x) \) as \( x \) approaches -1 from the right. Since \( f(x) = 2^x \) for \( x > -1 \), substitute \( x \) with values approaching -1: \[ \lim_{{x \to -1^+}} f(x) = 2^{-1} = \frac{1}{2}. \]

Step 4: Check for Discontinuity

A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value at that point all equal each other. Here, \( \lim_{{x \to -1^-}} f(x) = 2 \), but \( \lim_{{x \to -1^+}} f(x) = \frac{1}{2} \). Since these are not equal, the function is discontinuous at \( x = -1 \).

Step 5: Sketch the Graph

The graph should show a straight line \( x + 3 \) up to \( x = -1 \), ending with a solid dot at \((-1, 2)\). For \( x > -1 \), the graph will follow the curve \( 2^x \) starting from \((-1, 0.5)\) as an open circle, continuing to the right. This indicates a jump in the graph at \( x = -1 \).

Key Concepts

Piecewise Functions

Piecewise functions are an interesting type of function in mathematics, where two or more functions are pieced together to create a single function. The function is defined by different expressions depending on the input value.
These functions can be useful in real-world scenarios where a relationship behaves differently in different situations or intervals.
For example, when calculating costs, where a different pricing strategy applies for bulk purchases compared to smaller purchases.

• A typical piecewise function will have different expressions chosen based on intervals of the variable.
• The intervals might be defined using inequalities.
• Each piece of the function is usually separated by a specific point, where the behavior of the function changes.

Understanding piecewise functions involves determining which expression applies to each value of the variable. Therefore, knowing each condition is key. By carefully analyzing the different pieces and how they are defined, one can evaluate or sketch piecewise functions.

Limits and Continuity

The concepts of limits and continuity are central to understanding piecewise functions, especially where different parts of the function meet.
Limits allow us to understand how a function behaves as the input approaches a specific value, which is crucial for determining if a piecewise function is continuous at a particular point.
Continuity, on the other hand, ensures there are no abrupt changes, or jumps, in the function's graph.

• A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.
• If the left-hand limit (approaching the point from the left) and the right-hand limit (approaching the point from the right) both exist and are equal, the function is continuous at that point.
• When these conditions aren't met, the function may exhibit discontinuities like jumps, breaks, or oscillations at that point.

In our example, at the point where the piecewise function changes (\(x = -1\)), limits from the left and right differ, indicating a discontinuity. Understanding these concepts helps us discern why the function isn't smooth at this particular point.

Evaluating Limits

Evaluating limits involves finding out what a function approaches as the input nears a particular value. This process provides insight into the behavior of functions at crucial points, especially in piecewise functions where different rules apply depending on input ranges.
The technique of evaluating limits is fundamental when determining continuity or discontinuity.

• Calculating the left-hand limit involves substituting values slightly less than the point of interest into the function to see what value it approaches.
• Conversely, the right-hand limit is calculated using values slightly greater than the point in question.
• By comparing these two limits, we can make important assessments about continuity.

For instance, in the given piecewise function, calculating the limits from the left (\[x + 3\] as \(x\to -1^-\)) gives a result of 2, while the right-hand limit (\[2^x\] as \(x\to -1^+\)) differs, yielding \(\frac{1}{2}\). This difference in limits confirms the discontinuity at \(x = -1\). Understanding how to evaluate these limits is key to analyzing the behavior of complex functions like piecewise functions.