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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.1.31
Chapter 7, Problem 7.1.31

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)

Verified step by step guidance
1
Start by writing the function as an equation with y: \(y = \frac{x + 3}{x - 2}\).
To find the inverse function \(f^{-1}(x)\), swap the roles of \(x\) and \(y\): \(x = \frac{y + 3}{y - 2}\).
Solve this equation for \(y\) in terms of \(x\): multiply both sides by \((y - 2)\) to get \(x(y - 2) = y + 3\), then expand and rearrange terms to isolate \(y\).
Express \(y\) explicitly as a function of \(x\) to obtain \(f^{-1}(x)\).
Determine the domain and range of \(f^{-1}\) by considering the domain and range of the original function \(f\), and verify the inverse by checking that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

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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. 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 the equation 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 input values, while the range is the set of all possible output values. For inverse functions, the domain and range swap roles: the domain of f becomes the range of f⁻¹, and vice versa. Identifying these sets ensures the inverse is well-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 the identity function x. This step verifies the correctness of the inverse function and ensures no algebraic errors occurred.
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Related Practice
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) = x³ + 1

Textbook Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

39. lim (x → ∞) (ln 2x - ln(x + 1))

Textbook Question

Evaluate the integrals in Exercises 39–56.

41. ∫2y dy/(y²-25)

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Textbook Question

Evaluate the integrals in Exercises 39–56.

52. ∫(from π/4 to π/2)cot(t)dt

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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

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).