Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = (x + 3) / (x − 2)
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.27
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x³ + 1
Verified step by step guidance1
Start by writing the function given: \(f(x) = x^3 + 1\).
To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = x^3 + 1\).
Swap the roles of \(x\) and \(y\) to find the inverse: \(x = y^3 + 1\).
Solve this equation for \(y\) to express \(y\) in terms of \(x\): subtract 1 from both sides to get \(x - 1 = y^3\), then take the cube root to find \(y = \sqrt[3]{x - 1}\).
Identify the domain and range of \(f^{-1}(x)\). Since the original function \(f(x) = x^3 + 1\) has domain \((-\infty, \infty)\) and range \((-\infty, \infty)\), the inverse function will have domain \((-\infty, \infty)\) and range \((-\infty, \infty)\) as well.
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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 and f⁻¹(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y.
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Domain and Range of Functions and Their Inverses
The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs. For an inverse function, the domain and range swap roles compared to the original function. Identifying these sets ensures the inverse is properly defined.
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Verification of Inverse Functions
To confirm two functions are inverses, compose them in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). Both compositions should simplify to x, the identity function. This check validates the correctness of the inverse function found.
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Related Practice
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
37. y=s√(1-s²) + arccos(s)
Textbook Question
Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.
f(x) = (1 − x)³
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
30. lim (θ → 0) ((1/2)^θ - 1) / θ
1
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Textbook Question
Evaluate the integrals in Exercises 53–76.
53. ∫dx/√(9-x²)
Textbook Question
130. Use the identity arccot(u)=π/2 - arctan(u) to derive the formula for the derivative of arccot(u) in Table 7.4 from the formula for the derivative of arctan(u).