Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
67. ∫(from 0 to 2√3)dx/√(4+x²)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.42a
In Exercises 41–44:
a. Find f⁻¹(x).
42. f(x) = (x + 2) / (1 − x), a = 1/2
Verified step by step guidance1
Start by writing the function as an equation with y: \(y = \frac{x + 2}{1 - x}\).
To find the inverse function \(f^{-1}(x)\), swap the roles of \(x\) and \(y\): \(x = \frac{y + 2}{1 - y}\).
Multiply both sides of the equation by the denominator \((1 - y)\) to eliminate the fraction: \(x(1 - y) = y + 2\).
Distribute \(x\) on the left side: \(x - xy = y + 2\).
Group all terms involving \(y\) on one side and constants on the other: \(x - 2 = y + xy\). Then factor \(y\) out: \(x - 2 = y(1 + x)\). Finally, solve for \(y\): \(y = \frac{x - 2}{1 + x}\), which is \(f^{-1}(x)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. To find f⁻¹(x), solve the equation y = f(x) for x in terms of y, then interchange x and y. The inverse exists only if the function is one-to-one (bijective) on the domain considered.
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Inverse Cosine
Solving Rational Equations
Finding the inverse of a rational function often requires solving equations involving fractions. This involves clearing denominators, isolating variables, and simplifying expressions carefully to avoid extraneous solutions or domain issues.
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5:02Solving Logarithmic Equations
Domain and Range Considerations
When finding inverses, it is crucial to consider the domain of the original function and the range of the inverse. Restrictions on x (like denominators not being zero) affect the domain, and the inverse’s domain corresponds to the original function’s range.
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5:10Finding the Domain and Range of a Graph
Related Practice
Textbook Question
Suppose that the function g and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.
Assuming the inverse function g^(-1) is differentiable, find the slope of g^(-1)(x) at
a. x=1
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. a. sec^(-1)(-√2)
Textbook Question
78. Which one is correct, and which one is wrong? Give reasons for your answers.
a. lim (x → 0) (x² - 2x) / (x² - sin x) = lim (x → 0) (2x - 2) / (2x - cos x) = lim (x → 0) 2 / (2 + sin x) = 2 / (2 + 0) = 1
1
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Textbook Question
21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).
Textbook Question
77. Which one is correct, and which one is wrong? Give reasons for your answers.
a. lim (x → 3) (x - 3) / (x² - 3) = lim (x → 3) 1 / (2x) = 1/6