Problem 64 - Brief Applied Calculus

Question

For each piecewise linear function: a. Draw its graph (by hand or using a graphing calculator). b. Find the limits as \(x\) approaches 3 from the left and from the right. c. Is it continuous at \(x=3\) ? If not, indicate the first of the three conditions in the definition of continuity (page 87) that is violated. $$ f(x)=\left\\{\begin{array}{ll} 5-x & \text { if } x \leq 3 \\ x-1 & \text { if } x>3 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is continuous at \( x = 3 \).

Step 1: Understanding the Function

The function is piecewise, consisting of two linear segments. The first segment is \( f(x) = 5 - x \) for \( x \leq 3 \), and the second segment is \( f(x) = x - 1 \) for \( x > 3 \). We need to analyze these segments separately to understand the overall behavior of the function.

Step 2: Graph the Function

To graph the piecewise function, plot both segments on the same coordinate plane. For \( f(x) = 5 - x \), extend a line from \( x = -\infty \) to \( x = 3 \), including the point at \( x = 3 \). For \( f(x) = x - 1 \), start a new line just after \( x = 3 \) and extend to \( x = \infty \). The two segments meet at the boundary, \( x = 3 \), but may not connect.

Step 3: Calculate Limit from the Left

To find the limit as \( x \) approaches 3 from the left, evaluate the function \( f(x) = 5 - x \). As \( x \) approaches 3 from values less than 3, \( f(x) \) approaches \( 5 - 3 = 2 \). Hence, the left-hand limit is \( 2 \).

Step 4: Calculate Limit from the Right

To find the limit as \( x \) approaches 3 from the right, use the piece \( f(x) = x - 1 \). As \( x \) approaches 3 from values greater than 3, \( f(x) \) approaches \( 3 - 1 = 2 \). Thus, the right-hand limit is also \( 2 \).

Step 5: Evaluate Continuity at x = 3

For the function to be continuous at \( x = 3 \), it must meet three conditions: the function is defined at \( x = 3 \), both the left and right limits exist at \( x = 3 \), and the function value at \( x = 3 \) equals both limits. The value \( f(3) \) is defined as \( 5 - 3 = 2 \), and both limits as \( x \to 3 \) are 2. Thus, \( f(x) \) is continuous at \( x = 3 \).

Key Concepts

Limits

A limit helps us understand what value a function approaches as the input approaches a certain point. In the context of piecewise functions, this often involves examining each piece separately. For the exercise problem, the function divides at \( x = 3 \), which means the limit behavior should be checked from both sides using the appropriate segment of the piecewise function.

When approaching from the left (\( x \) values less than 3), we use the function piece \( f(x) = 5 - x \). As \( x \) gets closer to 3, \( f(x) \) approaches \( 2 \). This is known as the left-hand limit, written as \( \lim_{{x \to 3^-}} f(x) = 2 \).

From the right (\( x \) values greater than 3), we use the function piece \( f(x) = x - 1 \). As \( x \) approaches 3, \( f(x) \) also goes to \( 2 \). This is the right-hand limit, represented by \( \lim_{{x \to 3^+}} f(x) = 2 \).

Both limits match, indicating that the overall limit as \( x \to 3 \) is \( 2 \). This consistency in limits is a key element when considering continuity.

Continuity

A function is continuous at a point if it has no breaks, jumps, or holes at that point. Mathematically, to be continuous at \( x = 3 \), a piecewise function must satisfy three important conditions:

• The function must be defined at \( x = 3 \).
• The left-hand and right-hand limits as \( x \to 3 \) must exist.
• The value of the function at \( x = 3 \) should equal these limits.

Applying these conditions to our function:

* **Defined at \( x = 3 \):** For the segment \( x \leq 3 \), \( f(x) = 5 - x \). Thus, \( f(3) = 5 - 3 = 2 \).

* **Limits exist and equal \( f(3) \):** As discussed earlier, both the left-hand and right-hand limits are \( 2 \).

All conditions are met, so the function is continuous at \( x = 3 \). Continuity ensures no interruptions in the function's graph, making predictions or calculations at that point reliable and smooth.

Graphing Piecewise Functions

Graphing a piecewise linear function involves plotting each segment according to the defined rules and joining them at specified intervals. This method gives us a visual representation of how the function behaves across different intervals.

For our function, we have two case scenarios:

• **For \( x \leq 3 \):** The line follows \( f(x) = 5 - x \). Begin from \( x = -\infty \) up to \( x = 3 \). The endpoint \( (3, 2) \) is included since \( x \leq 3 \).
• **For \( x > 3 \):** The line is defined by \( f(x) = x - 1 \). It starts just after \( x = 3 \) and extends to \( x = \infty \). The function starts slightly past \( (3, 2) \) and continues upwards.

By correctly plotting each section on a graph, you'll notice the segments meet at \( x = 3 \) with no gaps or overlaps. This seamless joining visually confirms the continuity discussed earlier while offering a clear picture of the function’s overall behavior.