Problem 66 - Calculus Early Transcendentals

Question

Sketching graphs Sketch a possible graph of a function \(f\) that satisfies all the given conditions. Be sure to identify all vertical and horizontal asymptotes. $$\begin{aligned} &f(-1)=-2, f(1)=2, f(0)=0, \lim _{x \rightarrow \infty} f(x)=1\\\ &\lim _{x \rightarrow-\infty} f(x)=-1 \end{aligned}$$

Step-by-Step Solution

Answer

Question: Sketch a graph based on the given information about a function, f, which passes through the points (-1, -2), (0, 0), and (1, 2). As x approaches infinity, f(x) approaches 1, and as x approaches negative infinity, f(x) approaches -1. Answer: To sketch the graph, follow these steps: 1. Plot the points (-1, -2), (0, 0), and (1, 2). 2. Draw horizontal asymptotes at y = 1 and y = -1. 3. Connect the points with a smooth curve that approaches the asymptotes as x goes to infinity or negative infinity.

Step 1: Plot the given points

Plot the points (-1,-2), (0,0), and (1,2) on the Cartesian plane. These points form part of the function f.

Step 2: Determine horizontal asymptotes

Based on the given limits, we have two horizontal asymptotes: As \(x \rightarrow \infty\), \(f(x) \rightarrow 1\): This means there is a horizontal asymptote at y = 1 as x approaches infinity. As \(x \rightarrow -\infty\), \(f(x) \rightarrow -1\): This means there is a horizontal asymptote at y = -1 as x approaches negative infinity. Draw the horizontal asymptotes at y = 1 and y = -1.

Step 3: Sketch the possible graph

Now, we need to sketch a possible graph that passes through the given points and approaches the horizontal asymptotes. Since there is no information about the slopes or critical points, we can draw a smooth curve that goes through the points and slowly approaches the horizontal asymptotes as x goes to infinity or negative infinity. To ensure the graph satisfies all the given conditions, follow these guidelines: 1. Connect the points with a smooth curve. 2. As x approaches infinity, the curve should approach y=1 (without touching or crossing it). 3. As x approaches negative infinity, the curve should approach y=-1 (without touching or crossing it). Please note that there could be multiple graphs that satisfy these conditions, as the lack of information about slopes and critical points allows for various possible curve shapes.

Key Concepts

Horizontal Asymptotes

When sketching the graph of a function, one important aspect to consider is the horizontal asymptote. A horizontal asymptote represents a horizontal line that the graph of the function approaches as the input goes to either positive or negative infinity.
Although the graph gets very close to this line, it never actually touches or crosses it. It's like a boundary the function respects as it stretches outwards.

• As \(x \rightarrow \infty\), and if \(f(x)\rightarrow L\), there's a horizontal asymptote at \(y = L\).
• For example, if \(f(x)\rightarrow 1\) as \(x\rightarrow \infty\), then \(y = 1\) is a horizontal asymptote.

Given the exercise, we found that \(y=1\) and \(y=-1\) are horizontal asymptotes because:

• As \(x\rightarrow \infty\), \(f(x)\rightarrow 1\).
• As \(x\rightarrow -\infty\), \(f(x)\rightarrow -1\).

These indicate that our graph approaches these levels at the extremities, defining the long-term behavior of the function.

Vertical Asymptotes

Vertical asymptotes are another crucial part of graph sketching. Unlike horizontal ones, vertical asymptotes represent a vertical line where the function's value grows unbounded, often reaching positive or negative infinity.
This happens as the function nears a certain x-value. In many cases, this is due to division by zero or non-simplifiable logarithmic expressions at these points.

• The equation \(x = a\) is a vertical asymptote if \(f(x)\rightarrow \pm \infty\) as \(x\rightarrow a^+\) or \(x\rightarrow a^-\).

However, in the given exercise, no vertical asymptotes are explicitly mentioned. This typically suggests that there are no x-values where the function "blows up" or goes to infinity. Therefore, the focus should be on the points provided and the horizontal asymptotes when plotting the graph without additional vertical constraints.

Limits at Infinity

Understanding limits at infinity helps determine the behavior of a function as the input becomes very large or very small. It helps in identifying horizontal asymptotes and understanding the growth or decline rate of the function.
Imagine the graph of the function stretching its arms at infinity.

• \(\lim_{x\rightarrow \infty} f(x)\) indicates the direction the graph heads when \(x\) is very large.
• \(\lim_{x\rightarrow -\infty} f(x)\) tells what happens on the opposite end when \(x\) is very small.

In the exercise, we see the limits as follows:

• \(\lim_{x\rightarrow \infty} f(x) = 1\)
• \(\lim_{x\rightarrow -\infty} f(x) = -1\)

These limits directly lead us to place horizontal asymptotes at \(y = 1\) and \(y = -1\). Knowing limits at infinity greatly aids in sketching where the graph trends at its tail ends, seamlessly approaching its horizontal boundaries.