Decide which of the rational functions might have the given graph. 1 O A. f(x) = 1+ - *B. f(x) = 1-- O C. f(x) = 1-x 1 O D. f(x) = - 1

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Educational Content: Rational Function Graph Analysis**

**Graph Description:**
The graph displays a rational function. It has two distinct branches: one in the first quadrant and one in the third quadrant, both approaching but never touching the axes. 

- The vertical asymptote is at \( x = 0 \).
- The horizontal asymptote is at \( y = 1 \).

The graph suggests that as \( x \) approaches zero from the positive side, \( y \) increases without bound, and as \( x \) approaches zero from the negative side, \( y \) decreases without bound. As \( x \) moves towards positive or negative infinity, \( y \) approaches 1.

**Problem Statement:**
Decide which of the rational functions might have the given graph.

**Options:**

- **A.** \( f(x) = 1 + \frac{1}{x} \)
- **B.** \( f(x) = 1 - \frac{1}{x} \) (Correct Answer)
- **C.** \( f(x) = 1 - x \)
- **D.** \( f(x) = \frac{1}{x} - 1 \)

**Explanation:**
Option **B**, \( f(x) = 1 - \frac{1}{x} \), correctly models the behavior of the graph. The function has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 1 \), matching the observed characteristics.
Transcribed Image Text:**Educational Content: Rational Function Graph Analysis** **Graph Description:** The graph displays a rational function. It has two distinct branches: one in the first quadrant and one in the third quadrant, both approaching but never touching the axes. - The vertical asymptote is at \( x = 0 \). - The horizontal asymptote is at \( y = 1 \). The graph suggests that as \( x \) approaches zero from the positive side, \( y \) increases without bound, and as \( x \) approaches zero from the negative side, \( y \) decreases without bound. As \( x \) moves towards positive or negative infinity, \( y \) approaches 1. **Problem Statement:** Decide which of the rational functions might have the given graph. **Options:** - **A.** \( f(x) = 1 + \frac{1}{x} \) - **B.** \( f(x) = 1 - \frac{1}{x} \) (Correct Answer) - **C.** \( f(x) = 1 - x \) - **D.** \( f(x) = \frac{1}{x} - 1 \) **Explanation:** Option **B**, \( f(x) = 1 - \frac{1}{x} \), correctly models the behavior of the graph. The function has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 1 \), matching the observed characteristics.
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