Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 1 - Functions

Problema 1.19e

Capitolo 1, Problema 1.19e

Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .
Find the domain of ƒ o g.

Guida verificata passo dopo passo

Determine the domain of g(x) = $\sin$ x. Since $\sin$ x is defined for all real numbers, the domain of g(x) is all real numbers.

Consider the function f(x) = x^3. The function f(x) is a polynomial and is defined for all real numbers, so its domain is all real numbers.

To find the domain of the composite function (f $\circ$ g)(x) = f(g(x)), substitute g(x) into f(x) to get f($\sin$ x) = ($\sin$ x)^3.

Since $\sin$ x is defined for all real numbers and f(x) = x^3 is defined for all real numbers, the composite function f($\sin$ x) is also defined for all real numbers.

Therefore, the domain of the composite function f $\circ$ g is all real numbers.

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:

Concetti chiave

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

Composite Functions

A composite function is formed when one function is applied to the result of another function. In mathematical terms, if you have two functions f(x) and g(x), the composite function f(g(x)) is denoted as f o g. Understanding how to combine functions is essential for analyzing their behavior and determining their domains.

Video consigliato:

3:48

Evaluate Composite Functions - Special Cases

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For composite functions, the domain is influenced by both the inner function and the outer function. It is crucial to identify any restrictions, such as values that would lead to undefined expressions, to accurately determine the overall domain of the composite function.

Video consigliato:

Percorso guidato

5:10

Finding the Domain and Range of a Graph

Trigonometric Functions and Their Domains

Trigonometric functions, such as sine, have specific domains and ranges. The sine function, g(x) = sin(x), is defined for all real numbers, but when combined with other functions, its output must also fit within the domain of the outer function. In this case, since g(x) is the inner function for f(g(x)), understanding its output is vital for finding the domain of the composite function f o g.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Pratica correlata

Domanda del libro di testo

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~$\text{hr}$$ , which means its population is governed by the function $p$\left$(t$\right$)=150$\cdot{2^{\frac{t}{12}$}}$ , where $t$ is the number of hours after the first observation.

What is the population $4~$\text{days}$$ after the first observation?

321

views

Domanda del libro di testo

Identify the symmetry (if any) in the graphs of the following equations.

$y^2-4x^2=4$

415

views

Domanda del libro di testo

Taxicab fees A taxicab ride costs \$3.50 plus \$2.50 per mile. Let m be the distance (in miles) from the airport to a hotel. Find and graph the function c(m) that represents the cost of taking a taxi from the airport to the hotel. Also determine how much it will cost if the hotel is 9 miles from the airport.

312

views

Domanda del libro di testo

Decide whether $f$ , $g$ , or both represent one-to-one functions. <IMAGE>