Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of $$ g(x)=\left\\{\begin{aligned} -x+1 & \text { if } x<1 \\ x-1 & \text { if } 1

Step-by-Step Solution

Answer

Graph the three pieces for insights, then look at the limits and function values critically. Limits at \( x = 1 \) do not exist because left and right limits differ. \( g(1) \) is undefined. \( \lim_{x \to 2} g(x) \) and \( \lim_{x \to 2^+} g(x) \) ask for additional function insights.

Step 1: Analyze each piece

For \(x < 1\): \(g(x) = -x+1\) (line with slope \(-1\), passes through \((0,1)\), approaches \((1,0)\) from the left).
For \(1 < x < 2\): \(g(x) = x-1\) (line with slope 1, passes through \((1,0)\) to \((2,1)\), both endpoints open).
For \(x \geq 2\): \(g(x) = 5-x^2\) (downward parabola, \(g(2) = 1\), \(g(3) = -4\), etc.)

Step 2: Check continuity

At \(x=1\): left limit = 0, right limit = 0, but \(g(1)\) is undefined (open intervals). At \(x=2\): left limit = 1, \(g(2) = 1\), so continuous at \(x=2\).

Key Concepts

Limits and Continuity

When exploring limits and continuity in piecewise functions, it's crucial to understand how the function behaves at certain points, especially where the piecewise segments meet. In the given exercise, particularly for the limit as \( x \to 1 \) and \( x \to 2 \), we must evaluate how the limits from both sides (left-side and right-side limits) approach these points.
Remember, a limit \( \lim_{x \to c} f(x) \) exists if and only if the left-side limit \( \lim_{x \to c^-} f(x) \) and the right-side limit \( \lim_{x \to c^+} f(x) \) both exist and are equal. For example:

• For \( \lim_{x \to 1} g(x) \), the function's behavior is based on the intervals on either side of \( x=1 \). Examine \( g(x) = -x + 1 \) as \( x \to 1^- \) and \( g(x) = x - 1 \) as \( x \to 1^+ \). If these equal each other, the limit exists.
• Similarly, check \( \lim_{x \to 2} g(x) \) by considering left and right limits using the second and third expressions. If the values do not match, the limit does not exist.

Evaluating these limits helps to comprehend if the function is continuous or discontinuous at those critical points.
Continuity at a point \( x=c \) requires the function to be defined at \( c \), the limit as \( x \to c \) exists, and these equal \( f(c) \).
Thus, understanding limits and continuity for piecewise functions involves carefully analyzing each interval and how they connect at their end points.

Function Evaluation

Evaluating a piecewise function at a specific point requires careful examination of which interval the value belongs to. This involves identifying which expression from the piecewise function applies to the given \( x \)-value.
For example, when evaluating \( g(1) \) in our example, we need to determine which piecewise condition includes \( x=1 \). In this scenario, observe that the function segments on either side of \( x=1 \) are:

• \( g(x) = -x + 1 \) for \( x < 1 \), hence not applicable.
• \( g(x) = x - 1 \) for \( 1 < x < 2 \), also not applicable.

Here, since neither condition includes \( x=1 \), the function is not defined at \( x=1 \), and \( g(1) \) does not exist.
Function evaluation is straightforward for defined points. You simply plug the \( x \)-value into the corresponding expression from the piecewise function. It helps you understand if the function value is available or if there's discontinuity, aiding in exploring its overall nature.

Graph Sketching

Graph sketching of piecewise functions involves drawing each segment of the function according to its rule and paying attention to its domain. The goal is to depict how the function behaves in different intervals and highlight any discontinuities or critical points.
Start by plotting each piece of the given function:

• For \( g(x) = -x + 1 \), draw a line for the domain \( x < 1 \). This will be a descending line until it approaches but does not include \( x=1 \).
• Next, for \( 1 < x < 2 \), plot \( g(x) = x - 1 \). This line will rise within the interval, taking care not to include points where the function is not defined.
• Finally, sketch \( g(x) = 5 - x^2 \) for \( x \geq 2 \). This will make a downward-facing parabola from \( x=2 \) onwards including the point at \( x=2 \).

By connecting the dots and lines accordingly within their domains, you'll see the overall function shape. Be sure to note open and closed points based on whether the endpoint values are inclusive or exclusive.
Graph sketching serves as a visual representation, bringing the piecewise definition to life, and enables better understanding of the function's behavior across its different sections.

Step-by-Step Calculus Solutions

Solving calculus problems step-by-step provides clarity and a logical flow of reasoning. This technique is particularly important when dealing with piecewise functions and evaluating limits or continuity.
Approach each part methodically:

• Identify the piecewise function segments, recognizing the distinct interval for each expression. This foundational understanding assists with further calculations.
• For limits, examine each one separately using the left and right-hand perspectives as needed: calculate \( \lim_{x \to c^-} f(x) \) and \( \lim_{x \to c^+} f(x) \), then compare these findings.
• Evaluate the continuity by confirming if the function is defined, if the limit exists, and if it equates the function's value at specific points.
• Test for function evaluation at critical points by substituting values into the function according to its conditional expressions.

By systematically breaking the problem into smaller, manageable tasks, you can accurately solve each part, building confidence in dealing with more complex calculus concepts.
Detailed step-by-step analysis not only aids in solving individual problems but reinforces vital calculus principles, making future exercises more approachable and less daunting.