Also use graph to determine domain and range **Topic: Graphing Trigonometric Functions****Graph the following function:**Show at least two cycles. Use the graph to determine the domain and range of the function.\[ y = \csc \left( \frac{2\pi}{3} x \right) + 1 \]**Choose the correct graph below.**- **Option A:** This graph shows vertical asymptotes and csc function waves, ranging from \( -5 \) to \( 5 \), displaying two cycles. - **Option B:** This graph displays the csc function with vertical asymptotes and the wave structure is shifted up, ranging from \( -2 \) to \( 5 \) and horizontally scaled, covering \( -2\pi \) to \( 2\pi \). - **Option C:** This graph displays the csc function with vertical asymptotes and wave structure not appearing to match the given function scaling, ranging evenly along both axes.- **Option D:** This graph displays the csc function correctly reflecting \( y = \csc \left( \frac{2\pi}{3}x \right) + 1 \). It shows accurate vertical asymptotes and the wave structure fitting both the periodicity and vertical shift. It ranges from \( -3 \) to \( 5 \).**Use the graph to determine the domain of:** \[ y = \csc \left( \frac{2\pi}{3} x \right) + 1 \]**Explanation of Graph (Option D):**1. **Vertical Asymptotes:** These occur at \( x = \frac{3}{2}k\pi \) where \( k \) is an integer. The function has vertical asymptotes whenever \( \sin \left( \frac{2\pi}{3} x \right) = 0 \), which happens at \( \frac{2\pi}{3} x = k\pi \), resulting in \( x = \frac{3k}{2} \).2. **Range of Function:** The range of the function \( y = \csc \left( \frac{2\pi}{3} x \right) + 1 \) is \( (-\infty, 0) \cup (2, \infty) \).3. **Vertical Shift:** The \( +1 \) indicates an

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Graph the following function. Show at least two cycles. Use the graph to determine the domain and range of the function. y = csc 2x 3x +1 GREED Choose the correct graph below. OA. OB. OD. A 0 Use the graph to determine the domain of y = csc Ay 2T +1. O C. E CHIFLE HIFFILIHH HHHHHLE H TINTH HHINHI ER

Graph the following function. Show at least two cycles. Use the graph to determine the domain and range of the function. y = csc 2x 3x +1 GREED Choose the correct graph below. OA. OB. OD. A 0 Use the graph to determine the domain of y = csc Ay 2T +1. O C. E CHIFLE HIFFILIHH HHHHHLE H TINTH HHINHI ER

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

Also use graph to determine domain and range

**Topic: Graphing Trigonometric Functions**

**Graph the following function:**

Show at least two cycles. Use the graph to determine the domain and range of the function.

\[ y = \csc \left( \frac{2\pi}{3} x \right) + 1 \]

**Choose the correct graph below.**

- **Option A:** This graph shows vertical asymptotes and csc function waves, ranging from \( -5 \) to \( 5 \), displaying two cycles.

- **Option B:** This graph displays the csc function with vertical asymptotes and the wave structure is shifted up, ranging from \( -2 \) to \( 5 \) and horizontally scaled, covering \( -2\pi \) to \( 2\pi \).

- **Option C:** This graph displays the csc function with vertical asymptotes and wave structure not appearing to match the given function scaling, ranging evenly along both axes.

- **Option D:** This graph displays the csc function correctly reflecting \( y = \csc \left( \frac{2\pi}{3}x \right) + 1 \). It shows accurate vertical asymptotes and the wave structure fitting both the periodicity and vertical shift. It ranges from \( -3 \) to \( 5 \).

**Use the graph to determine the domain of:**

\[ y = \csc \left( \frac{2\pi}{3} x \right) + 1 \]

**Explanation of Graph (Option D):**
1. **Vertical Asymptotes:** These occur at \( x = \frac{3}{2}k\pi \) where \( k \) is an integer. The function has vertical asymptotes whenever \( \sin \left( \frac{2\pi}{3} x \right) = 0 \), which happens at \( \frac{2\pi}{3} x = k\pi \), resulting in \( x = \frac{3k}{2} \).

2. **Range of Function:** The range of the function \( y = \csc \left( \frac{2\pi}{3} x \right) + 1 \) is \( (-\infty, 0) \cup (2, \infty) \).

3. **Vertical Shift:** The \( +1 \) indicates an

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