Problem 28 $$\text { Sketch the graphs of (... [FREE SOLUTION]

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Calculus: One and Several Variables

Saturnino L. Salas, Garret J. Etgen, Einar Hille

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 7: Problem 28

$$\text { Sketch the graphs of (a) } y=\cosh ^{-1} x,(\mathrm{b}) y=\tanh ^{-1} x$$

Short Answer

Expert verified

The graph of $y= \cosh^{-1} x$ starts at the point (1, 0) and increases to infinity as we move to the right, with a slope that gets less negative. The graph of $y=\tanh^{-1} x$ crosses the origin and has vertical asymptotes at $x = -1$ and $x = 1$. As $x$ tends to 1 from the left, $y$ tends to infinity, and as $x$ tends to -1 from the right, $y$ tends to -infinity.

Step by step solution

Sketch the graph of $y= \cosh^{-1} x$

To sketch the graph of $y= \cosh^{-1} x$, we first note that the function is defined for $x \geq 1$. The graph starts at the point (1, 0); as $x$ tends to infinity, $y$ also tends toward infinity. Since $\cosh^{-1} x$ is a decreasing function, the slope of the graph is negative. So put the starting point at (1,0) and draw a curve that increases as you move to the right, making sure to indicate that the curve is tending to infinity as $x$ gets large. The curve never touches the $x$-axis.

Sketch the graph of $y= \tanh^{-1} x$

To sketch the graph of $y=\tanh^{-1} x$, we note that the function is defined for $-1 < x < 1$. As $x$ tends to 1 from the left, $y$ tends to infinity, and as $x$ tends to -1 from the right, $y$ tends to -infinity. At $x = 0$, $y = 0$. So, we can say that the graph crosses the origin and increases as we move to the right and decreases as we move to the left, with asymptotes at $x = -1$ and $x = 1$. Draw a point at (0,0) and curve that goes up as you move to the right and goes down as you move to the left. The curve becomes parallel to $y$-axis near $x = 1$ and $x = -1$. These lines, $x = 1$ and $x = -1$, are the vertical asymptotes for the graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Sketching

Graph sketching is an essential skill in understanding the behavior of mathematical functions visually. This involves plotting various points and understanding the general shape of the graph based on the function's definition and derived properties.

Start by identifying the domain of the function. This is the set of all possible input values (x-values) for which the function is defined.
Next, look for key points, such as intercepts with the x-axis and y-axis, where the function changes direction (turning points), and where the slope is zero.
Find asymptotes, which are lines that the graph approaches but never touches. Vertical asymptotes are particularly important in functions like inverse hyperbolic functions.

By following these steps, you'll create a visual representation that reflects the function's behavior accurately.

Cosh Inverse

The hyperbolic cosine inverse function, denoted as $ y = \cosh^{-1} x $, is the inverse of the hyperbolic cosine function $ x = \cosh y $. It is defined only for $ x \geq 1 $. The reason is that the hyperbolic cosine, $ \cosh x $, only outputs values greater than or equal to 1.

When sketching $ y = \cosh^{-1} x $:

Start at (1,0) since it is the minimum value for $ x $.
The graph increases monotonically; as $ x $ increases, $ y $ also increases towards infinity.
The curve is always above the x-axis, never touching it because $ \cosh^{-1} x $ doesn't reach zero or any negative numbers.

The function is always increasing, meaning it consistently slopes upwards as $ x $ increases. Understanding this behavior helps in identifying its upward curving shape.

Tanh Inverse

The hyperbolic tangent inverse function, represented as $ y = \tanh^{-1} x $, is a fascinating exponential function. Its domain is $-1 < x < 1$, which means it's only defined between these x-values. The hyperbolic tangent function, conversely, compresses infinitely between -1 and 1, so the inverse extends between these bounds.

To sketch the graph of $ y = \tanh^{-1} x $:

Note that it crosses the origin at the point (0, 0).
It increases and approaches infinity as $ x $ approaches 1 from the left side.
Simultaneously, as $ x $ approaches -1 from the right side, $ y $ trends towards negative infinity.
The graph has vertical asymptotes at $ x = 1 $ and $ x = -1 $. The curvature appears to tighten as it nears these bounds.

These asymptotes make the shape of the $ y = \tanh^{-1} x $ graph distinct and easy to recognize. The function鈥檚 smooth and continuous transition through the origin, along with its asymptotes, make up its unique characteristics.

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91影视

$$\text { Sketch the graphs of (a) } y=\cosh ^{-1} x,(\mathrm{b}) y=\tanh ^{-1} x$$

Short Answer

Step by step solution

Sketch the graph of \(y= \cosh^{-1} x\)

Sketch the graph of \(y= \tanh^{-1} x\)

Key Concepts

Graph Sketching

Cosh Inverse

Tanh Inverse

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