Chapter 5: Q. 7 (page 348)

In Problems 5–7, find f o g, g o f, f o f, and g o g for each pair of functions. State the domain of each composite function.
$f (x) = \frac{x + 1}{x - 1} g (x) = \frac{1}{x}$

Short Answer

Expert verified

$(f o g) (x) = \frac{1 + x}{1 - x} D o m a i n : {x | x \neq 0, x \neq 1} (g o f) (x) = \frac{x - 1}{x + 1} D o m a i n : {x | x \neq - 1, x \neq 1} (f o f) (x) = x d o m a i n : {x | x \neq 1} (g o g) (x) = x d o m a i n : {x | x \neq 0}$

Step by step solution

Given information

we are given with two functions

$f (x) = \frac{x + 1}{x - 1} g (x) = \frac{1}{x}$

Part(a) : Step 1:Composition and domain

$f (x) = \frac{x + 1}{x - 1} g (x) = \frac{1}{x} (f o g) (x) = f (g (x)) = f (\frac{1}{x}) = \frac{\frac{1}{x} + 1}{\frac{1}{x} - 1} = \frac{1 + x}{1 - x} D o m a i n " {x | x \neq 0, x \neq 1}$

Part(b): Step 1 :Composition and domain

$f (x) = \frac{x + 1}{x - 1} g (x) = \frac{1}{x} (g o f) (x) = g (f (x)) = g (\frac{x + 1}{x - 1}) = \frac{1}{\frac{x + 1}{x - 1}} = \frac{x + 1}{x - 1} D o m a i n " {x | x \neq - 1, x \neq 1}$

Part(c) : Step 1:Composition and domain

$f (x) = \frac{x + 1}{x - 1} g (x) = \frac{1}{x} (f o f) (x) = f (f (x)) = f (\frac{x + 1}{x - 1}) = \frac{\frac{x + 1}{x - 1} + 1}{\frac{x + 1}{x - 1} - 1} = x D o m a i n : {x | x \neq 1}$

Part(d) : Step 1: Composition and domain

$f (x) = \frac{x + 1}{x - 1} g (x) = \frac{1}{x} (g o g) (x) = g (g (x)) = g (\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x D o m a i n : {x | x \neq 0}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 5–7, find f o g, g o f, f o f, and g o g for each pair of functions. State the domain of each composite function.
$f (x) = \frac{x + 1}{x - 1} g (x) = \frac{1}{x}$

Short Answer

Step by step solution

Given information

Part(a) : Step 1:Composition and domain

Part(b): Step 1 :Composition and domain

Part(c) : Step 1:Composition and domain

Part(d) : Step 1: Composition and domain

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Pure Maths

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Problems 5–7, find f o g, g o f, f o f, and g o g for each pair of functions. State the domain of each composite function. f(x)=x+1x-1g(x)=1x

Short Answer

Step by step solution

Given information

Part(a) : Step 1:Composition and domain

Part(b): Step 1 :Composition and domain

Part(c) : Step 1:Composition and domain

Part(d) : Step 1: Composition and domain

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Pure Maths

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Problems 5–7, find f o g, g o f, f o f, and g o g for each pair of functions. State the domain of each composite function.
$f (x) = \frac{x + 1}{x - 1} g (x) = \frac{1}{x}$