Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problema 43d

Capitolo 3, Problema 43d

Find f/g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

Guida verificata passo dopo passo

Step 1: Understand the problem. You are tasked with finding the quotient of two functions, f(x) and g(x), which is represented as (f/g)(x) = f(x)/g(x). Additionally, you need to determine the domain of the resulting function.

Step 2: Write the given functions. f(x) = (5x + 1)/(x² - 9) and g(x) = (4x - 2)/(x² - 9). To find (f/g)(x), divide f(x) by g(x): (f/g)(x) = [f(x)] / [g(x)].

Step 3: Simplify the division. Dividing two fractions involves multiplying the numerator of the first fraction by the reciprocal of the second fraction. So, (f/g)(x) = [(5x + 1)/(x² - 9)] * [(x² - 9)/(4x - 2)].

Step 4: Cancel out common factors. Notice that (x² - 9) appears in both the numerator and denominator, so it cancels out, leaving (f/g)(x) = (5x + 1)/(4x - 2).

Step 5: Determine the domain. The domain of a function is the set of all x-values for which the function is defined. For (f/g)(x), the denominator of both f(x) and g(x) must not be zero, and g(x) itself must not be zero. Solve x² - 9 = 0 to find x-values that make the denominator zero, and solve 4x - 2 = 0 to find x-values that make g(x) zero. Exclude these values from the domain.

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:

11m

Concetti chiave

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

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this case, f(x) and g(x) are both rational functions, where the numerator and denominator are polynomials. Understanding how to manipulate and simplify these functions is crucial for finding their quotient and determining their domains.

Video consigliato:

6:04

Intro to Rational Functions

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero, as division by zero is undefined. Identifying these restrictions is essential for determining the valid inputs for f/g.

Video consigliato:

3:51

Domain Restrictions of Composed Functions

Quotient of Functions

The quotient of two functions, denoted as f/g, is calculated by dividing the output of function f by the output of function g. This operation requires careful attention to the domains of both functions to ensure that the resulting function is defined. Additionally, simplifying the quotient may involve factoring and canceling common terms in the numerator and denominator.

Video consigliato:

3:49

Product, Quotient, and Power Rules of Logs

Pratica correlata

Domanda del libro di testo

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x², g(x) = x² - 2

157

views

Domanda del libro di testo

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x − 1)², x ≤ 1

637

views

Domanda del libro di testo

Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = (3x+1)/(x² - 25), g(x) = (2x -4)/(x² - 25)

800

views

Domanda del libro di testo

Find f−g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

1196

views

Domanda del libro di testo

In Exercises 41–44, use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-3, 6) and perpendicular to the line whose equation is y = (1/3)x + 4

216

views

Domanda del libro di testo

Find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

1290

views