Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.1.31
Capitolo 7, Problema 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)

Guida verificata passo dopo passo
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 \(f^{-1}(f(x)) = x\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
4:49
Inverse Cosine

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.
Video consigliato:
Percorso guidato
5:10
Finding the Domain and Range of a Graph

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.
Video consigliato:
4:49
Inverse Cosine
Pratica correlata
Domanda del libro di testo

41. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room where the temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions.

a. How much longer would it take the soup to cool to 35°C?

5
views
Domanda del libro di testo

Evaluate the integrals in Exercises 33–54.

∫ 2t e^(-t²) dt

20
views
Domanda del libro di testo

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

85. y = log₂(8t^(ln 2))

40
views
Domanda del libro di testo

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

83. y = 3^(log₂ t)

27
views
Domanda del libro di testo

In Exercises 7–10, determine from its graph if the function is one-to-one.

f(x) = 3 - x, x < 0

= 3, x ≥ 0

32
views
Domanda del libro di testo

Evaluate the integrals in Exercises 39–56.

52. ∫(from π/4 to π/2)cot(t)dt

33
views