/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 108 The hyperbolic tangent is define... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 108

The hyperbolic tangent is defined by tanh \(x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\) Find its inverse function \(\tanh ^{-1} x\).

Short Answer

Expert verified
The inverse function is \(\tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{x + 1}{1 - x}\right)\).

Step by step solution

01

Define the Equation

Start by stating the equation for the hyperbolic tangent: \[ y = \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \] We need to find the inverse, \(\tanh^{-1}(x)\), such that \(x = \tanh^{-1}(y)\).
02

Swap Variables

To find the inverse function, we swap the variables \(x\) and \(y\), resulting in:\[ x = \frac{e^y - e^{-y}}{e^y + e^{-y}} \] This represents \(y\) in terms of \(x\).
03

Solve for Exponential Terms

Multiply both sides by \(e^y + e^{-y}\) to eliminate the fraction:\[ x(e^y + e^{-y}) = e^y - e^{-y} \] Simplify this to:\[ xe^y + xe^{-y} = e^y - e^{-y} \]
04

Rearrange to Consolidate Terms

Bring like terms to one side:\[ xe^y - e^y = - xe^{-y} - e^{-y} \] Factor out \(e^y\) on the left and \(e^{-y}\) on the right:\[ (x - 1)e^y = -(x + 1)e^{-y} \]
05

Express in Terms of One Exponential

Multiply the whole equation by \(e^y\) to express everything in terms of \(e^{2y}\):\[ (x - 1)e^{2y} = -(x + 1) \] Now isolate \(e^{2y}\):\[ e^{2y} = \frac{-(x + 1)}{x - 1} \]
06

Solve for \(y\)

Take the natural logarithm on both sides to solve for \(2y\):\[ 2y = \ln\left(\frac{x + 1}{1 - x}\right) \] Finally, solve for \(y\) by dividing both sides by 2:\[ y = \frac{1}{2} \ln\left(\frac{x + 1}{1 - x}\right) \] Thus, the inverse function is \(\tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{x + 1}{1 - x}\right)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Hyperbolic Tangent
The hyperbolic tangent function, denoted as \( \tanh(x) \), represents a relationship in hyperbolic geometry similar to the tangent function in trigonometry. This function is defined by:
  • \( \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \)
Here, \( e \) is the base of the natural logarithm, which is approximately 2.718.

The hyperbolic tangent function is used to map real numbers within the range -1 to 1, making it very useful in various fields like engineering and physics.
A few properties of the hyperbolic tangent function include:
  • Range: Values between -1 and 1.
  • Odd Function: Since \( \tanh(-x) = -\tanh(x) \).
  • Smoothness: It is a continuous and smooth curve.
These properties make the hyperbolic tangent function ideal for mathematical modeling, especially in neural network algorithms where non-linear activation functions are vital.
Inverse Functions
Inverse functions reverse the roles of inputs and outputs. For the function \( y = \tanh(x) \), the inverse is finding \( x \) given \( y \). This means solving \( y = \frac{e^x - e^{-x}}{e^x + e^{-x}} \) for \( x \).

To find the inverse function of \( \tanh(x) \):
  • First, swap \( x \) and \( y \) in the equation to express \( x \) in terms of \( y \).
  • Rearrange to solve for \( y \) by eliminating fractions and combining like terms.
The result derives the inverse function:
  • \( \tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{x + 1}{1 - x}\right) \)
This process shows how an inverse function gives us the original input from an output, which is crucial for reverting transformations in equations and calculating original data values.
Exponential Functions
Exponential functions involve the constant, \( e \), raised to the power of a variable and are expressed as \( e^x \). These functions describe growth or decay processes, which is a fundamental concept in areas such as biology, finance, and physics.

In the context of the hyperbolic tangent, exponential components \( e^x \) and \( e^{-x} \) appear in the numerator and denominator:
  • \( \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \)
Solving the inverse of the hyperbolic tangent involves multiple steps using exponential equations:
  • Eliminate the fraction involving \( e^x \) and \( e^{-x} \) by multiplying both sides.
  • Rewriting terms using \( e^{2y} \) transforms the problem into a solvable logarithmic form.
The manipulation of exponential functions in this context highlights their versatility in solving complex mathematical expressions and obtaining the inverse of a hyperbolic function.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Wing Shan just graduated from dental school owing \(\$ 80,000\) in student loans. The annual interest is \(6 \% .\) Her time \(t\) to pay off the loan is given by $$t=-\frac{\ln \left[1-\frac{80,000(0.06)}{n R}\right]}{n \ln \left(1+\frac{0.06}{n}\right)}$$ where \(n\) is the number of payment periods per year and \(R\) is the periodic payment. a. Use a graphing utility to graph $$t_{1}=-\frac{\ln \left[1-\frac{80,000(0.06)}{12 x}\right]}{12 \ln \left(1+\frac{0.06}{12}\right)} \text { as } Y_{1} \text { and }$$ $$t_{2}=-\frac{\ln \left[1-\frac{80,000(0.06)}{26 x}\right]}{26 \ln \left(1+\frac{0.06}{26}\right)} \text { as } Y_{2}$$ Explain the difference in the two graphs. b. Use the \([\text { TRACE }]\) key to estimate the number of years that it will take Wing Shan to pay off her student loan if she can afford a monthly payment of \(\$ 800\) c. If she can make a biweekly payment of \(\$ 400\), estimate the number of years that it will take her to pay off the loan. d. If she adds \(\$ 200\) more to her monthly or \(\$ 100\) more to her biweekly payment, estimate the number of years that it will take her to pay off the loan.

Explain the mistake that is made. State the domain of the logarithmic function \(f(x)=\ln |x|\) in interval notation. Solution: since the absolute value eliminates all negative numbers, the domain is the set of all real numbers. Interval notation: \((-\infty, \infty)\) This is incorrect. What went wrong?

If \(\$ 9,000\) is invested in a savings account earning \(6 \%\) interest compounded continuously, how many years will pass until there is \(\$ 15,000 ?\)

In calculus we prove that the derivative of \(f+g\) is \(f^{\prime}+g^{\prime}\) and that the derivative of \(f-g\) is \(f^{\prime}-g^{\prime} .\) It is also shown in calculus that if \(f(x)=\ln x\) then \(f^{\prime}(x)=\frac{1}{x}\) Use these properties to find the derivative of \(f(x)=\ln \frac{1}{x}\)

The New York Philharmonic has a sound intensity of \(100 \mathrm{dB}\). How many watts per square meter does the orchestra emit?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.