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f(x) 10 6 4 2 -27 2л 2. 2, 3,

f(x) 10 6 4 2 -27 2л 2. 2, 3,

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Find an equation for f(x) using the cosecant function.

f(x) =
The image displays a graph of the function \( f(x) = \sec{x} + \csc{x} \). This trigonometric function exhibits periodic behavior with certain key characteristics. ### Key Features of the Graph: 1. **Vertical Asymptotes**: - The graph contains vertical asymptotes at \( x = -\pi, \, x = 0, \, x = \pi \), and \( x = 2\pi \). These are indicated with dashed orange lines. The function approaches positive or negative infinity as \( x \) approaches these values. 2. **Behavior and Range**: - The function shows a periodic and repeating pattern due to the properties of the secant (\(\sec{x}\)) and cosecant (\(\csc{x}\)) functions. - Between each pair of asymptotes, the function has a U-shape above the x-axis and an upside-down U-shape below the x-axis. - The peaks occur just before reaching the asymptotes. 3. **Intervals Displayed**: - The graph covers the interval from \( x = -2\pi \) to \( x = 2\pi \). 4. **Symmetry**: - The function demonstrates even symmetry, indicated by the similar shapes and positions of the curves within each periodic section. ### Explanation: - The combination of \(\sec{x} + \csc{x}\) results in singularities (where the function is undefined) due to the nature of the individual trigonometric components. Both \(\sec{x}\) and \(\csc{x}\) tend to infinity when \(\cos{x}\) or \(\sin{x}\) reach zero. This graph helps visualize the complex, periodic, and discontinuous nature of the function \( \sec{x} + \csc{x} \), aiding in the understanding of its properties within the bounds of a single period.
Transcribed Image Text:The image displays a graph of the function \( f(x) = \sec{x} + \csc{x} \). This trigonometric function exhibits periodic behavior with certain key characteristics. ### Key Features of the Graph: 1. **Vertical Asymptotes**: - The graph contains vertical asymptotes at \( x = -\pi, \, x = 0, \, x = \pi \), and \( x = 2\pi \). These are indicated with dashed orange lines. The function approaches positive or negative infinity as \( x \) approaches these values. 2. **Behavior and Range**: - The function shows a periodic and repeating pattern due to the properties of the secant (\(\sec{x}\)) and cosecant (\(\csc{x}\)) functions. - Between each pair of asymptotes, the function has a U-shape above the x-axis and an upside-down U-shape below the x-axis. - The peaks occur just before reaching the asymptotes. 3. **Intervals Displayed**: - The graph covers the interval from \( x = -2\pi \) to \( x = 2\pi \). 4. **Symmetry**: - The function demonstrates even symmetry, indicated by the similar shapes and positions of the curves within each periodic section. ### Explanation: - The combination of \(\sec{x} + \csc{x}\) results in singularities (where the function is undefined) due to the nature of the individual trigonometric components. Both \(\sec{x}\) and \(\csc{x}\) tend to infinity when \(\cos{x}\) or \(\sin{x}\) reach zero. This graph helps visualize the complex, periodic, and discontinuous nature of the function \( \sec{x} + \csc{x} \), aiding in the understanding of its properties within the bounds of a single period.
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