Question

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

Short answer

Question: Sketch a graph of a function \(f(x)\) such that \(f(-1) = -2\), \(f(1) = 2\), \(f(0) = 0\), \(\lim_{x \rightarrow \infty} f(x) = 1\), and \(\lim_{x \rightarrow-\infty} f(x) = -1\) and mention its asymptotes. Answer: The graph of the function \(f(x)\) would include the points \((-1, -2)\), \((1, 2)\), and \((0, 0)\), with horizontal asymptotes at \(y = 1\) and \(y = -1\). There are no vertical asymptotes mentioned in the problem. The function should approach the asymptotes as \(x\) goes to \(\infty\) and \(-\infty\).

Step-by-step solution

Identify the Given Points

We are given these points: - \(f(-1) = -2\) - \(f(1) = 2\) - \(f(0) = 0\) Make sure to include these points in the graph of the function.

Determine the Horizontal Asymptotes

We need to find the horizontal asymptotes. We are given these limits: - \(\lim _{x \rightarrow \infty} f(x)=1\) - \(\lim _{x \rightarrow-\infty} f(x)=-1\) The function approaches 1 as \(x\) goes to \(\infty\) and approaches -1 as \(x\) goes to \(-\infty\). Therefore, we have two horizontal asymptotes: - \(y=1\) - \(y=-1\)

Find Vertical Asymptotes

There are no vertical asymptotes mentioned in this problem. Therefore, we don't need to find any vertical asymptotes.

Sketch the Graph

Now, using the given points and asymptotes, sketch the graph of the function \(f(x)\): 1. Plot the points \((-1, -2)\), \((1, 2)\), and \((0, 0)\) on the graph. 2. Draw the horizontal asymptotes \(y=1\) and \(y=-1\) as dashed lines. 3. Make sure the function approaches the asymptotes as \(x\) goes to \(\infty\) and \(-\infty\). 4. Since there are no vertical asymptotes, ensure the function does not have any abrupt vertical jumps. Remember that the graph is a sketch and may not be a smooth curve or a combination of simpler functions. The focus should be on fulfilling the given conditions.

Key concepts

Understanding Horizontal Asymptotes

Horizontal asymptotes are an important feature to consider when sketching a function's graph. They indicate the behavior of a function as the input values become very large (approaching infinity) or very small (approaching negative infinity). In simple terms, a horizontal asymptote represents a horizontal line that the graph of a function gets closer to as it extends towards infinity in either direction.

In our exercise, the horizontal asymptotes are determined by the following limits:

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

These limits tell us that the function approaches the line \(y = 1\) as \(x\) goes to infinity, and approaches \(y = -1\) as \(x\) goes towards negative infinity. This means that the graph will flatten out, becoming nearly level with these horizontal lines as it moves to the extreme sides of the graph.

Exploring Limits

Limits help describe the behavior of a function as it approaches certain points or as the input values increase or decrease without bound. They are crucial in defining horizontal asymptotes and understanding end behavior of functions.

In our example, the limits are given as:

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

These limits simply mean that as \(x\) increases to large positive values, \(f(x)\) becomes closer to 1, and as \(x\) decreases to large negative values, \(f(x)\) becomes closer to -1. Limits are particularly helpful for identifying asymptotic behavior and predicting how a graph behaves outside of plotted points.

Analyzing Function Behavior

Function behavior involves how a function acts across its domain, including patterns and trends exhibited in a graph. This encompasses discussions of increasing/decreasing intervals, extreme points, and any asymptotic behavior.

In our task to sketch the function, you should consider the noted horizontal asymptotes along with the given points. Since there are no vertical asymptotes, the function does not have any sudden jumps or undefined points.

• The function should smoothly approach the horizontal lines at \(y=1\) and \(y=-1\) as \(x\) extends to positive and negative infinity respectively.
• The points \((-1, -2)\), \((0, 0)\), and \((1, 2)\) act as guides to help shape the curve between the asymptotic behaviors.

Such understanding helps not only in plotting the graph correctly but also in anticipating the trends a function might have as more data points are considered.

Interpreting Given Points

Given points are specific coordinates that the function must pass through. They provide crucial information for positioning the curve accurately on the graph.

In our exercise, the key points provided are:

• \((-1, -2)\)
• \((0, 0)\)
• \((1, 2)\)

These coordinates guide where the function actually touches or crosses the graph. Understanding these points helps in ensuring that the function's graph accurately reflects the required conditions. Additionally, they serve as a focal reference to ensure that the overall shape coincides with the expected behavior connected to horizontal asymptotes and overall trends of the function.