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 1

Capitolo 3, Problema 1

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) = 4x and g(x) = x/4

Guida verificata passo dopo passo

First, find the composition \( f(g(x)) \). This means you substitute \( g(x) \) into \( f(x) \). Since \( f(x) = 4x \) and \( g(x) = \frac{x}{4} \), write \( f(g(x)) = f\left( \frac{x}{4} \right) \).

Next, evaluate \( f\left( \frac{x}{4} \right) \) by replacing the \( x \) in \( f(x) = 4x \) with \( \frac{x}{4} \). This gives \( f\left( \frac{x}{4} \right) = 4 \times \frac{x}{4} \).

Simplify the expression \( 4 \times \frac{x}{4} \) by canceling the 4 in numerator and denominator, resulting in \( f(g(x)) = x \).

Now, find the composition \( g(f(x)) \). Substitute \( f(x) \) into \( g(x) \). Since \( g(x) = \frac{x}{4} \) and \( f(x) = 4x \), write \( g(f(x)) = g(4x) \).

Evaluate \( g(4x) \) by replacing \( x \) in \( g(x) = \frac{x}{4} \) with \( 4x \), giving \( g(4x) = \frac{4x}{4} \). Simplify this to get \( g(f(x)) = x \). Since both compositions \( f(g(x)) \) and \( g(f(x)) \) equal \( x \), the functions \( f \) and \( g \) are inverses of each other.

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.

Function Composition

Function composition involves applying one function to the result of another, denoted as f(g(x)) or g(f(x)). It means substituting the entire function g(x) into f(x), or vice versa, to create a new function. Understanding this helps evaluate combined functions and analyze their behavior.

Video consigliato:

4:56

Function Composition

Inverse Functions

Inverse functions reverse the effect of each other, so applying one after the other returns the original input: f(g(x)) = x and g(f(x)) = x. To determine if two functions are inverses, their compositions must simplify to the identity function, meaning the output equals the input.

Video consigliato:

4:30

Graphing Logarithmic Functions

Linear Functions and Their Properties

Linear functions have the form f(x) = mx + b, where m and b are constants. In this question, both functions are linear with no constant term, making it easier to analyze their compositions and inverses. Recognizing linearity simplifies calculations and understanding of function behavior.

Video consigliato:

5:36

Change of Base Property

Pratica correlata

Domanda del libro di testo

Determine whether each relation is a funciton, Give the domain and range for each relation. (1, 10), (2, 500), (13, π)

144

views

Domanda del libro di testo

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. (4, 7) and (8, 10)

views

Domanda del libro di testo

Determine whether each relation is a function. Give the domain and range for each relation.{(1, 2), (3, 4), (5, 5)}

465

views

Domanda del libro di testo

Find the domain of each function. f(x) = 2(x+5)

887

views

Domanda del libro di testo

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

g(x) = f(x)+1