Skip to main content
Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 2 - Functions and GraphsProblema 10
Capitolo 3, Problema 10

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = ∛(x − 4) and g(x) = x³ +4

Guida verificata passo dopo passo
1
First, recall that the composition of functions f(g(x)) means substituting g(x) into f(x). So, write down f(g(x)) as f(g(x)) = \(\sqrt\)[3]{g(x) - 4}.
Next, substitute g(x) = x^3 + 4 into the expression for f(g(x)), giving f(g(x)) = \(\sqrt\)[3]{(x^3 + 4) - 4}.
Simplify the expression inside the cube root: (x^3 + 4) - 4 simplifies to x^3, so f(g(x)) = \(\sqrt\)[3]{x^3}.
Now, recall that the cube root of x cubed is just x, so f(g(x)) simplifies to x.
Next, find g(f(x)) by substituting f(x) into g(x). Write g(f(x)) = (f(x))^3 + 4, then substitute f(x) = \(\sqrt\)[3]{x - 4} to get g(f(x)) = (\(\sqrt\)[3]{x - 4})^3 + 4. Simplify this expression to check if it equals 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:
1m

Concetti chiave

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

Function Composition

Function composition involves applying one function to the result of another, denoted as f(g(x)) or g(f(x)). It requires substituting the entire expression of one function into the variable of the other, allowing us to analyze how functions combine and transform inputs.
Video consigliato:
4:56
Function Composition

Inverse Functions

Inverse functions reverse the effect of each other, meaning f(g(x)) = x and g(f(x)) = x for all x in the domains. Identifying inverses involves checking if composing the functions in both orders returns the original input, confirming they undo each other's operations.
Video consigliato:
4:30
Graphing Logarithmic Functions

Cube Roots and Cubes

The cube root function, ∛x, and the cube function, x³, are inverse operations. Understanding how these functions interact, especially with shifts like (x - 4) or +4, is essential to correctly compose and verify if two functions are inverses.
Video consigliato:
03:41
Special Products - Cube Formulas
Pratica correlata
Domanda del libro di testo

Evaluate each function at the given values of the independent variable and simplify. (a) f(-2), (b) f(1), (c) f(2)

114
views
Domanda del libro di testo

Use the graph of y = f(x) to graph each function g.

g(x) = ½ f(x)

1721
views
Domanda del libro di testo

Use the graph of y = f(x) to graph each function g.

g(x) = f(-x)+3

1044
views
Domanda del libro di testo

The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = x +3

635
views
Domanda del libro di testo

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = 2, passing through (3, 5)

86
views
Domanda del libro di testo

Find the domain of each function. f(x) = 1/(x+7) + 3/(x-9)

1256
views