Problem 48 - Calculus: Early Transcendentals

Question

Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. $$ \begin{array}{l} h(-1)=2, \lim _{x \rightarrow-1^{-}} h(x)=0, \lim _{x \rightarrow-1^{+}} h(x)=3 \\ h(1)=\lim _{x \rightarrow 1^{-}} h(x)=1, \lim _{x \rightarrow 1^{+}} h(x)=4 \end{array} $$

Step-by-Step Solution

Answer

Question: Based on the given limits and specified values of h(x), sketch the graph of the function h(x). Important features: - h(-1) = 2 - $\lim _{x \rightarrow-1^{-}} h(x) = 0$ - $\lim _{x \rightarrow-1^{+}} h(x) = 3$ - h(1) = 1 - $\lim _{x \rightarrow 1^{-}} h(x) = 1$ - $\lim _{x \rightarrow 1^{+}} h(x) = 4$ Rough sketch: 1. Mark points (-1,2) and (1,1) on the graph. 2. Draw a curve descending to 0 as x approaches -1 from the left but doesn't touch the x-axis. 3. Draw a curve starting from y=3 at x=-1 and descends to meet the point (-1,2). 4. Draw the graph to approach and meet the point (1,1) from the left. 5. Draw a curve starting from y=4 at x=1 and descends to meet the graph at the point (1,1).

Step 1: Identify important features

First, let's write down the important features given by the exercise: $$\begin{array}{l}h(-1)=2, \lim _{x \rightarrow-1^{-}} h(x)=0, \lim _{x \rightarrow-1^{+}} h(x)=3 \\\ h(1)=\lim _{x \rightarrow 1^{-}} h(x)=1, \lim _{x \rightarrow 1^{+}} h(x)=4 \end{array}$$ We can draw a few conclusions from this information: - At x = -1, the function takes the value 2. - As x approaches -1 from the left, the function converges to 0. - As x approaches -1 from the right, the function converges to 3. - At x = 1, the function takes the value 1 and its left limit also converges to 1. - As x approaches 1 from the right, the function converges to 4.

Step 2: Sketch the function

Based on the given properties, let's try to sketch a graph for the function h(x). 1. Mark the points (-1, 2) and (1, 1) on the graph as these are the values of the function at x = -1 and x = 1. 2. Since the limit of h(x) is 0 approaching x = -1 from the left, draw a curve descending to 0 as x approaches -1 from the left but doesn't touch the x-axis. 3. Then, since the limit as x approaches -1 from the right is 3, draw another curve that starts from the y-value of 3 at x = -1 and descends to meet the point (-1, 2). 4. The given information tells us that the limit as x approaches 1 from the left is equal to h(1), so the graph should approach and meet the point (1, 1) from the left. 5. The limit as x approaches 1 from the right is 4, so draw a curve that starts from the y-value of 4 at x = 1 and descends to meet the graph at the point (1,1). After sketching these curves, we should have a rough sketch of the graph for the function h(x). Note that this graph will not be unique, as we can always modify the graph between the points provided it satisfies the given properties.

Key Concepts

Function Limits

When we talk about the limits of a function, we're referring to what value the function gets close to as the input approaches a certain point. This concept is crucial when sketching graphs because it guides us on how the graph behaves near key points. Let's break it down with the help of our example:

• At \(x = -1\), we have two limits: one as \(x\) approaches from the left, and one from the right.
• As \(x\) approaches \(-1\) from the left, \(h(x)\) gets closer to 0. This means that the graph will approach, but not necessarily touch, the y-value of 0 as it nears \(x = -1\) from the left side.
• On the other hand, as \(x\) approaches \(-1\) from the right, \(h(x)\) heads towards 3. This indicates a jump at \(x = -1\), since the values coming from either side do not meet.
• For \(x = 1\), the function converges to the same value from the left as the function value itself, \(h(x) = 1\).
• As \(x\) approaches 1 from the right, the function limit is 4, showing another jump in value.

These limits tell us about potential discontinuities and help guide our sketch of the graph.

Discontinuities in Functions

Discontinuities occur when a function jumps, breaks, or has holes in its graph. Understanding discontinuities is key to drawing accurate graphs. In our case, we see different types of discontinuities:

• **Jump Discontinuities:** At \(x = -1\) and \(x = 1\), there are jump discontinuities. A jump occurs when the limits from both sides are finite but unequal. For \(x = -1\), the left limit is 0 and the right limit is 3, indicating a jump. Similarly, for \(x = 1\), the left limit and the value \(h(1)\) are both 1, but the right limit is 4.
• **Removable Discontinuities:** These happen when a 'hole' can theoretically be filled by adjusting the function value. In our situation, there are no removable discontinuities since we're focused on understanding how the function behaves around specific points and accepting the given points of discontinuity.
• **Infinite Discontinuities:** These are not present in this function, since the limits are finite around the discussed points.

These discontinuities significantly affect how we sketch the graph, showing where the function fails to be continuous.

Graphical Analysis of Functions

Graphical analysis allows us to visualize the behavior of functions step by step, which is especially useful when dealing with complex functions or incomplete information. By analyzing the graph, we can:

• **Locate Points:** Knowing specific values, like \(h(-1) = 2\) and \(h(1) = 1\), lets us plot these distinct points precisely on our graph.
• **Follow Trends:** The behavior as shown by the function limits as \(x\) approaches \(-1\) and \(1\) gives us a guide to draw the curves on either side of these points. From the left and right, the function's direction changes, indicating these trends.
• **Identify Gaps and Jumps:** The discontinuities highlight where jumps happen in the graph. Understanding these interruptions helps us not expect a smooth continuation across these points.

By combining these insights, you can produce a clear and insightful graph. While this exercise doesn't require a single specific graph, it trains you to sketch with the guidance of function properties and behavior patterns, ensuring a deeper understanding.