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)
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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 7.1.41a
In Exercises 41–44:
a. Find f⁻¹(x).
41. f(x) = 2x + 3, a = −1
Verified step by step guidance1
Start with the given function: \(f(x) = 2x + 3\).
To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = 2x + 3\).
Swap the variables \(x\) and \(y\) to get \(x = 2y + 3\); this step reflects the idea that the inverse function reverses the roles of inputs and outputs.
Solve the equation \(x = 2y + 3\) for \(y\): subtract 3 from both sides to get \(x - 3 = 2y\), then divide both sides by 2 to isolate \(y\): \(y = \frac{x - 3}{2}\).
Rewrite \(y\) as \(f^{-1}(x)\) to express the inverse function: \(f^{-1}(x) = \frac{x - 3}{2}\).
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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. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x. Finding the inverse involves solving the equation y = f(x) for x in terms of y.
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Solving Linear Equations
To find the inverse of a linear function like f(x) = 2x + 3, you solve for x by isolating it on one side of the equation. This involves algebraic manipulation such as subtracting constants and dividing by coefficients to express x as a function of y.
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07:17Linearization
Domain and Range Considerations
When finding an inverse, it's important to consider the domain and range of the original function and its inverse. For linear functions, the domain and range are typically all real numbers, but specifying points like a = -1 helps evaluate the inverse at particular values.
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5:10Finding the Domain and Range of a Graph
Related Practice
Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
42. f(x) = (x + 2) / (1 − x), a = 1/2
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Textbook Question
145. The linearization of eˣ at x = 0
a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.
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Textbook Question
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. 10x^4 + 30x + 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).
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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
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