Problem 47 - 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. $$\begin{array}{l}g(1)=0, g(2)=1, g(3)=-2, \lim _{x \rightarrow 2} g(x)=0 \\\\\lim _{x \rightarrow 3^{-}} g(x)=-1, \lim _{x \rightarrow 3^{+}} g(x)=-2\end{array}$$

Step-by-Step Solution

Answer

Answer: Around x=2, the function g(x) approaches 0, and around x=3, there seems to be a discontinuity with a jump from -1 to -2 as x crosses 3. The function behaves differently as it approaches x=3 from the left side and the right side.

Step 1: Plot given points

First, let's note down the points given: $$ g(1) = 0, \; g(2) = 1, \; g(3) = -2 $$ Plot these points on the x-y plane.

Step 2: Show the limit behavior at x=2 and x=3

We have the following limits: $$ \lim_{x\rightarrow 2} g(x) = 0 \\ \lim_{x\rightarrow 3^-} g(x) = -1 \\ \lim_{x\rightarrow 3^+} g(x) = -2 $$ As x approaches 2, the function must go towards 0. As x approaches 3 from the left, the function goes to -1, and as x approaches 3 from the right, the function goes to -2.

Step 3: Connect the points to form the graph

Now, let's connect the points plotted in step 1 while taking into account the limit behavior at x=2 and x=3. 1. Start by connecting the points (1, 0) and (2, 1) using a straight line, to show the \(\lim_{x\rightarrow 2} g(x) = 0\) behavior. 2. Draw a hole at point (3, -1) since this is the limit as x approaches 3 from the left. 3. Connect (2, 1) to the hole at (3, -1) with a straight line. 4. From the hole at (3, -1), start a line that goes to the point (3, -2). This represents the jump from -1 to -2 in the function when x crosses 3. 5. Connect the point (3, -2) onwards. You can continue drawing according to the context or stop as desired. With these steps completed, the graph of the function g(x) will match the given points and limit behavior.

Key Concepts

Plotting Points on Coordinate Plane

Before we delve into the complexities of functions, it's crucial to master plotting points on the coordinate plane. This skill is akin to placing pins on a map, allowing us to visually display the information provided by a function at specific inputs, or x-values.

Begin with a two-dimensional grid, one axis for the x-values (horizontal) and another for the y-values (vertical). Each point consists of two numbers corresponding to its x and y coordinates. For example, the point (1, 0) means you move 1 unit right along the x-axis and 0 units on the y-axis, placing your 'pin' directly on the x-axis. Similarly, (3, -2) indicates 3 units right and 2 units down into the negative territory of the y-axis.

Plotting the given points, such as \(g(1) = 0, g(2) = 1, g(3) = -2\), forms the skeleton of our graph. These 'bones' guide us as we draw our graph, indicating where our function must pass through. Understanding this initial mapping is essential before venturing into the function's intricacies, like limits, continuity, and discontinuity.

Understanding Limits of Functions

Limits are a foundational component in grasping the behavior of functions, particularly when they approach specific points. Imagine you're hiking towards a mountain peak. As you get closer, you can predict the direction the path will lead, even if the peak is shrouded in mist. Similarly, limits predict the value that a function approaches as the input, or x, nears a certain number.

When we see \(\lim_{x\rightarrow 2} g(x) = 0\), it tells us that as x gets infinitesimally close to 2, g(x) approaches the value 0. However, we must differentiate between approaching from the left (\(x \rightarrow a^{-}\)) or from the right (\(x \rightarrow a^{+}\)). Our exercise specifies \(\lim_{x\rightarrow 3^{-}} g(x) = -1\) and \(\lim_{x\rightarrow 3^{+}} g(x) = -2\), which means the function approaches different values depending on the direction of approach.

Graphically, we interpret these limits by drawing our function's curve towards the predicted value as x nears the specified point. Sometimes we may encounter a hole or jump in the graph, indicators of the fascinating concept of discontinuity.

Continuity and Discontinuity of Functions

Continuity in a function resembles a smooth drawing line, unbroken as it courses through the coordinate plane. Discontinuity, on the other hand, represents interruptions where the function 'leaps' or is undefined at certain points.

In our case, the presence of different limits as x approaches 3 from either side signals a discontinuous point. Specifically, the gap between \(\lim_{x\rightarrow 3^{-}} g(x) = -1\) and \(g(3) = -2\) creates a 'hole' in our graph and a jump from -1 to -2 on the y-axis. We visualize this by leaving an open circle at the point where the function cannot take that value and drawing a solid dot where the function actually continues.

Discontinuity often has various types, including point discontinuity as seen here, or other forms like jump discontinuity and asymptotic behavior. Recognizing and plotting these on a graph enhances our understanding of how functions can act in unexpected manners, revealing the complexities of their overall behavior.