Skip to main content
Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 1.89
Capitolo 1, Problema 1.89

Finding all inverses Find all the inverses associated with the following functions, and state their domains.


ƒ(x) = 2 / ( x² + 2)

Guida verificata passo dopo passo
1
The given function is \( f(x) = \frac{2}{x^2 + 2} \). This is a rational function where the numerator is a constant and the denominator is a quadratic expression.
To find the inverse, we need to solve for \( x \) in terms of \( y \). Start by setting \( y = \frac{2}{x^2 + 2} \).
Rearrange the equation to express \( x^2 + 2 \) in terms of \( y \): \( y(x^2 + 2) = 2 \). Then, solve for \( x^2 \): \( x^2 = \frac{2}{y} - 2 \).
Take the square root of both sides to solve for \( x \): \( x = \pm \sqrt{\frac{2}{y} - 2} \).
The expression \( \frac{2}{y} - 2 \) must be non-negative for the square root to be defined, so \( \frac{2}{y} \geq 2 \), which implies \( y \leq 1 \). Therefore, the domain of the inverse function is \( y \in (-\infty, 1] \).

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:
6m

Concetti chiave

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

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.
Video consigliato:
4:49
Inverse Cosine

Domain and Range

The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce. When finding the inverse of a function, the domain of the original function becomes the range of the inverse, and vice versa.
Video consigliato:
Percorso guidato
5:10
Finding the Domain and Range of a Graph

Finding Inverses Algebraically

To find the inverse of a function algebraically, you typically start by replacing f(x) with y, then swap x and y in the equation. After that, you solve for y to express it in terms of x. This new expression represents the inverse function, and it is important to also determine its domain based on the original function.
Video consigliato:
4:49
Inverse Cosine
Pratica correlata
Domanda del libro di testo

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.


logb x/y

307
views
Domanda del libro di testo

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.


G o G

237
views
Domanda del libro di testo

Sketch the graph of the inverse of ƒ. <IMAGE>

387
views
Domanda del libro di testo

Find the inverse f−1(x)f^{-1}\(\left\)(x\(\right\)) of each function (on the given interval, if specified).

f(x)=2x2+1f\(\left\)(x\(\right\))=\(\frac{2}{x^2+1}\), for x≥0x\(\geq{0}\)

417
views
Domanda del libro di testo

Solving equations Solve the following equations.


ln x= -1

334
views
Domanda del libro di testo

The National Weather Service releases approximately 70,00070,000 radiosondes every year to collect data from the atmosphere. Attached to a balloon, a radiosonde rises at about 10001000 ft/min until the balloon bursts in the upper atmosphere. Suppose a radiosonde is released from a point 66 ft above the ground and that 55 seconds later, it is 8383 ft above the ground. Let f(t)f\(\left\)(t\(\right\)) represent the height (in feet) that the radiosonde is above the ground tt seconds after it is released. Evaluate f(5)−f(0)5−0\(\frac{f\left(5\right)-f\left(0\right)}{5-0}\) and interpret the meaning of this quotient.

514
views