The-Organic-Chemistry-Tutor
This video covers the graphs of inverse hyperbolic functions, including hyperbolic tangent, cotangent, cosecant, and secant. The graphs are obtained by reflecting the original graphs across the line y=x. The domains and ranges of the regular and inverse functions are related, with the domain becoming the range and vice versa. The speaker provides examples and explains the asymptotes and restrictions for each inverse function, as well as their usefulness in solving problems. The importance of comparing regular and inverse graphs to understand their properties and relationships is emphasized.
In this section, the instructor explains how to graph the inverse hyperbolic functions by reflecting the original graph across the line Y equals X. The domain of the regular hyperbolic function becomes the range of the inverse function, and vice versa. The instructor provides examples of the regular hyperbolic sine, cosine, and tangent functions and their corresponding inverse functions. The domain and range of the inverse functions are determined based on the domain and range of the regular functions.
In this section, the video explains how to draw the graph of the inverse hyperbolic functions for hyperbolic tangent, hyperbolic cosecant, and hyperbolic secant. They describe how the horizontal asymptotes become vertical asymptotes for the inverse functions and how the X and Y values are switched. They also discuss the domains and ranges of the regular and inverse hyperbolic functions, and how the graph of inverse secant is restricted to the upper half of the graph.
In this section, we learn about the graphs of inverse hyperbolic functions. The first function discussed is the hyperbolic tangent, which is flipped when reflected across the line y=x. The domain and range are switched, but the domain is the same (from 0 to 1). The range of the inverse function is from 0 to infinity. The second function discussed is the hyperbolic cotangent, which has two horizontal and one vertical asymptote. Its inverse has two vertical asymptotes at x=-1 and x=1 and a horizontal asymptote at y=0. The domain of the inverse function is from negative infinity to negative one union one to infinity, and the range is from negative infinity to zero union zero to infinity.
In this section, the speaker concludes their discussion on the graphs of the inverse hyperbolic functions. They highlight that the inverse hyperbolic functions are useful for solving problems involving hyperbolic functions, especially those that involve the natural logarithm. Additionally, the speaker emphasizes the importance of comparing the graphs of the hyperbolic functions and their inverses to get a better understanding of their properties and the relationships between them.
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