**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.

In the graph given, it can be observed that- At x =1, function f(x)=0 As x→ +infinity, function f(x)→1 x→ - infinity, function f(x)→1 x→ 0+, function f(x)→-infinity x→ 0-, function f(x)→+ infinityAll the above observation is fulfilled by- F(x)=1-1/X So, option (b) is correct

Answered: Decide which of the rational functions…

Math

Calculus

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.

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