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)=−2f\left(-1\right)=-2, f(1)=2f\left(1\right)=2, f(0)=0f\left(0\right)=0, lim⁡x→∞f(x)=1{\displaystyle\lim_{x\to\infty}{f(x)=1}}, lim⁡x→−∞f(x)=−1{\displaystyle\lim_{x\to-\infty}{f(x)=-1}}

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.5.86

Chapter 2, Problem 2.5.86

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

Verified step by step guidance

Identify the given points on the graph: The function passes through the points (-1, -2), (1, 2), and (0, 0). Plot these points on the coordinate plane.

Determine the horizontal asymptotes: As x approaches infinity, f(x) approaches 1, indicating a horizontal asymptote at y = 1. As x approaches negative infinity, f(x) approaches -1, indicating a horizontal asymptote at y = -1.

Consider the behavior near the given points: Since f(-1) = -2, f(1) = 2, and f(0) = 0, the function must pass through these points smoothly, possibly indicating a polynomial-like behavior between these points.

Sketch the graph considering the asymptotic behavior: As x approaches infinity, the graph should approach y = 1 without crossing it. Similarly, as x approaches negative infinity, the graph should approach y = -1.

Ensure continuity and smoothness: Connect the plotted points with a smooth curve that respects the asymptotic behavior and passes through the given points, ensuring the graph is continuous and differentiable where applicable.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where a function approaches infinity, typically at points where the function is undefined. Horizontal asymptotes indicate the behavior of a function as x approaches infinity or negative infinity, showing the value the function approaches. Understanding asymptotes is crucial for sketching the overall behavior of a function.

Limits

Limits describe the behavior of a function as the input approaches a certain value. They are fundamental in calculus for defining continuity, derivatives, and integrals. In this context, limits at infinity help determine the horizontal asymptotes of the function, indicating the values the function approaches as x becomes very large or very small. Mastery of limits is essential for analyzing function behavior.

Function Values

Function values at specific points provide critical information about the graph of a function. For instance, knowing that f(-1) = -2, f(0) = 0, and f(1) = 2 allows us to plot key points on the graph. These values help in understanding the function's behavior in different intervals and are essential for sketching the graph accurately. They also assist in identifying local maxima, minima, and points of intersection.

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