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. $$\begin{aligned}&p(0)=2, \lim _{x \rightarrow 0} p(x)=0, \lim _{x \rightarrow 2} p(x) \text { does not exist. }\\\&p(2)=\lim _{x \rightarrow 2^{+}} p(x)=1\end{aligned}$$

Short answer

Based on the given information about the function \(p(x)\) and using the four steps above, the graph should show the following features: 1. A point at (0, 2). 2. A curve that approaches the x-axis as x approaches 0. 3. A vertical asymptote or discontinuity at x = 2, indicated by a dashed vertical line. 4. The curve gradually approaching the point (2, 1) from the right side. Please note that without a specific formula, the graph may have different shapes, but it must include these key features to represent the given properties of the function \(p(x)\).

Step-by-step solution

Plot the point (0, 2)

Since the function passes through the point \((0, 2)\), we start by plotting the point \((0, 2)\) on the graph.

Indicate the limit as x approaches 0

Given \(\lim_{x\rightarrow 0} p(x) = 0\), we need to show that as x approaches 0, the value of the function approaches 0. We can draw a curve approaching the x-axis as it reaches 0.

Indicate a discontinuity at x = 2

Since \(\lim_{x\rightarrow 2} p(x)\) does not exist, there must be a discontinuity or a vertical asymptote at \(x = 2\). Draw a dashed vertical line at \(x = 2\) to indicate this.

Show the limit as x approaches 2 from the right

The function satisfies \(p(2)= \lim_{x\rightarrow 2^{+}} p(x) = 1\). Draw the curve approaching the point \((2, 1)\) from the right side. Now, combine the information from Steps 1-4 to sketch the graph of the function \(p(x)\).

Key concepts

Limit of a Function

Limits are a fundamental concept to understand when sketching graphs. The limit of a function describes the value that the output of a function approaches as the input approaches a certain point.
For the given exercise, we are particularly interested in what happens as \(x\) approaches 0.

If \(\lim_{x \rightarrow 0} p(x) = 0\), this means that as \(x\) gets closer and closer to 0, the output \(p(x)\) gets closer and closer to the value 0.
This can occur even if the function value at that point is different. In this case, while \(p(0) = 2\), the function's output gets very close to 0 as \(x\) approaches 0, showing that limits can differ from actual function values.

Understanding limits is important because it helps identify the behavior of functions around points where direct calculation or observation is difficult. By plotting a graph, you can visualize how the function behaves near a point.

Discontinuity

Discontinuity in a function graph is a place where the function does not behave as expected. There are various types of discontinuities, but the most significant in this exercise is when the limit does not exist.

• In mathematical terms, when you say \(\lim_{x \rightarrow 2} p(x)\) does not exist, it indicates a discontinuity at \(x = 2\).

This could mean multiple behaviors such as a jump, vertical asymptote, or an infinite limit.

The original step-by-step solution suggests that when drawing the graph, we indicate this discontinuity with a dashed vertical line at \(x=2\).
This visual cue helps understand that there's a break in the function, where the function may not be defined, or it jumps abruptly to a new value.
Recognizing discontinuities helps in understanding the overall behavior and nature of a function's graph.

Function Properties

Understanding the properties of a function greatly aids in graph sketching and analysis. A function property refers to characteristics such as limits, values, and behaviors at certain points.

In the exercise, we're given specifics like \(p(0) = 2\), \(\lim_{x \rightarrow 2^+} p(x) = 1\), and other limits directly illuminating the function's properties:

• \(p(0) = 2\): Indicates that the graph definitely passes through the point (0, 2).
• \(\lim_{x \rightarrow 2^+} p(x) = 1\): Shows that from the right of \(x = 2\), the function approaches but might not reach \(p(x) = 1\).

By synthesizing the function properties with graphs, you achieve a fuller understanding of what the behavior of \(p(x)\) looks like in context.

These properties help create a more accurate depiction of the function by marking important points and anticipated behaviors on the graph.