**Mathematics - Asymptotes of Rational Functions****Topic: Asymptotes of the function $ f(x) = \frac{x - 2}{x^2 - 4} $**---Let $ f $ be the function defined by $ f(x) = \frac{x - 2}{x^2 - 4} $ for all real numbers $ x $ for which $ f(x) $ is a real number.---**Question:**Which of the following are equations of the asymptotes of the graph of $ f $ in the xy-plane?---**Options (Select all that apply):**- [ ] $ x = -2 $- [ ] $ x = 0 $- [ ] $ x = 2 $- [ ] $ y = -2 $- [ ] $ y = 0 $- [ ] $ y = 2 $---**Analysis of the Function:**1. **Identifying Vertical Asymptotes:** Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. The denominator $x^2 - 4$ factors to $(x + 2)(x - 2)$, so the function is undefined at $ x = 2 $ and $ x = -2 $. Evaluating the numerator, we see that it does not become zero at these points, so $ x = 2 $ and $ x = -2 $ are vertical asymptotes.2. **Identifying Horizontal Asymptotes:** To find horizontal asymptotes for a rational function $ \frac{P(x)}{Q(x)} $, consider the degrees of $ P(x) $ and $ Q(x) $. Here both the numerator $ (x - 2) $ and the denominator $ (x^2 - 4) $ are polynomials. Since the degree of the denominator is higher (quadratic) than the numerator (linear), $ y = 0 $ is the horizontal asymptote.---**Correct Answers:**- [x] $ x = -2 $- [x] $ x = 2 $- [x] $ y = 0 $**Explanation of Graphs/Diagrams:**Not applicable for this transcription, as there are no graphs or diagrams

x-2 for all real numbers x for which f(x) is a real number. 7-4 Let f be the function defined by f(x) = %3D Which of the following are equations of the asymptotes of the graph of f in the xy-plane? Select all that apply. O1= -2 Ox= 2 Oy=-2 O y = 0 Oy 2

x-2 for all real numbers x for which f(x) is a real number. 7-4 Let f be the function defined by f(x) = %3D Which of the following are equations of the asymptotes of the graph of f in the xy-plane? Select all that apply. O1= -2 Ox= 2 Oy=-2 O y = 0 Oy 2

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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Question

**Mathematics - Asymptotes of Rational Functions**

**Topic: Asymptotes of the function $ f(x) = \frac{x - 2}{x^2 - 4} $**

---

Let $ f $ be the function defined by $ f(x) = \frac{x - 2}{x^2 - 4} $ for all real numbers $ x $ for which $ f(x) $ is a real number.

---

**Question:**
Which of the following are equations of the asymptotes of the graph of $ f $ in the xy-plane?

---

**Options (Select all that apply):**
- [ ] $ x = -2 $
- [ ] $ x = 0 $
- [ ] $ x = 2 $
- [ ] $ y = -2 $
- [ ] $ y = 0 $
- [ ] $ y = 2 $

---

**Analysis of the Function:**

1. **Identifying Vertical Asymptotes:**
Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. The denominator $x^2 - 4$ factors to $(x + 2)(x - 2)$, so the function is undefined at $ x = 2 $ and $ x = -2 $.
Evaluating the numerator, we see that it does not become zero at these points, so $ x = 2 $ and $ x = -2 $ are vertical asymptotes.

2. **Identifying Horizontal Asymptotes:**
To find horizontal asymptotes for a rational function $ \frac{P(x)}{Q(x)} $, consider the degrees of $ P(x) $ and $ Q(x) $. Here both the numerator $ (x - 2) $ and the denominator $ (x^2 - 4) $ are polynomials. Since the degree of the denominator is higher (quadratic) than the numerator (linear), $ y = 0 $ is the horizontal asymptote.

---

**Correct Answers:**
- [x] $ x = -2 $
- [x] $ x = 2 $
- [x] $ y = 0 $

**Explanation of Graphs/Diagrams:**
Not applicable for this transcription, as there are no graphs or diagrams

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