Q. 5 Write out each of the following ... [FREE SOLUTION]

Chapter 4: Q. 5 (page 384)

Write out each of the following functions as a piecewise-defined function that does not involve absolute values.
$a) f (x) = |x| b) f (x) = |2 - 3 x| c) f (x) = |x^{2} - 3 x - 4| d) f (x) = ||x - 1| - 1|$

Short Answer

Expert verified

$Part (a) : f (x) = \{\begin{cases} - x & ; x < 0 \\ x & ; x 鈮� 0 \end{cases} Part (b) : f (x) = \{\begin{cases} 3 x - 2 & ; x > \frac{2}{3} \\ 2 x - 3 & ; x 鈮� \frac{2}{3} \end{cases} Part (c) : 鈬� f (x) = \{\begin{cases} - (x^{2} - 3 x - 4) & ; - 1 < x < 4 \\ x^{2} - 3 x - 4 & ; else \end{cases} Part (d) : f (x) = \{\begin{cases} - x & ; x < 0 \\ x & ; 0 < x < 1 \\ 2 - x & ; 1 鈮� x < 2 \\ x - 2 & ; x 鈮� 2 \end{cases}$

Step by step solution

Part (a) Step 1. Given information is:

$f (x) = | x |$

Part (a) Step 2. Piecewise Defined Function

$Consider : f (x) = | x | 鈬� f (x) = \{\begin{cases} - x & ; x < 0 \\ x & ; x 鈮� 0 \end{cases}$

Part (b) Step 1. Given information is:

$f (x) = |2 - 3 x|$

Part (b) Step 2. Piecewise Defined Function

$Consider : f (x) = | 2 - 3 x | f (x) = 3 x - 2; 2 - 3 x < 0 x > \frac{2}{3} f (x) = 2 - 3 x; 2 - 3 x 鈮� 0 x 鈮� \frac{2}{3} 鈬� f (x) = \{\begin{cases} 3 x - 2 & ; x > \frac{2}{3} \\ 2 x - 3 & ; x 鈮� \frac{2}{3} \end{cases}$

Part (c) Step 1. Given information is:

$f (x) = |x^{2} - 3 x - 4|$

Part (c) Step 2. Piecewise Defined Function

$Consider : f (x) = | x^{2} - 3 x - 4 | f (x) = - (x^{2} - 3 x - 4); x^{2} - 3 x - 4 < 0 f (x) = x^{2} - 3 x - 4; x^{2} - 3 x - 4 鈮� 0; - 1 < x < 4 鈬� f (x) = \{\begin{cases} - (x^{2} - 3 x - 4) & ; - 1 < x < 4 \\ x^{2} - 3 x - 4 & ; else \end{cases}$

Part (d) Step 1. Given information is:

$f (x) = ||x - 1| - 1|$

Part (d) Step 2. Piecewise Defined Function

$Consider : f (x) = ||x - 1| - 1| Let g (x) = |x - 1| therefore, g (x) = \{\begin{cases} 1 - x & ; x < 1 \\ x - 1 & ; x 鈮� 1 \end{cases} f (x) = - (1 - x - 1) = x; x > 0, x < 1 f (x) = (1 - x - 1) = - x; x < 0 f (x) = - (x - 1 - 1) = 2 - x; x < 2, x 鈮� 1 f (x) = (x - 1 - 1) = x - 2; x 鈮� 2 鈬� f (x) = \{\begin{cases} - x & ; x < 0 \\ x & ; 0 < x < 1 \\ 2 - x & ; 1 鈮� x < 2 \\ x - 2 & ; x 鈮� 2 \end{cases}$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Write out each of the following functions as a piecewise-defined function that does not involve absolute values.
$a) f (x) = |x| b) f (x) = |2 - 3 x| c) f (x) = |x^{2} - 3 x - 4| d) f (x) = ||x - 1| - 1|$

Short Answer

Step by step solution

Part (a) Step 1. Given information is:

Part (a) Step 2. Piecewise Defined Function

Part (b) Step 1. Given information is:

Part (b) Step 2. Piecewise Defined Function

Part (c) Step 1. Given information is:

Part (c) Step 2. Piecewise Defined Function

Part (d) Step 1. Given information is:

Part (d) Step 2. Piecewise Defined Function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Write out each of the following functions as a piecewise-defined function that does not involve absolute values.a)f(x)=xb)f(x)=2-3xc)f(x)=x2-3x-4d)f(x)=x-1-1

Short Answer

Step by step solution

Part (a) Step 1. Given information is:

Part (a) Step 2. Piecewise Defined Function

Part (b) Step 1. Given information is:

Part (b) Step 2. Piecewise Defined Function

Part (c) Step 1. Given information is:

Part (c) Step 2. Piecewise Defined Function

Part (d) Step 1. Given information is:

Part (d) Step 2. Piecewise Defined Function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Write out each of the following functions as a piecewise-defined function that does not involve absolute values.
$a) f (x) = |x| b) f (x) = |2 - 3 x| c) f (x) = |x^{2} - 3 x - 4| d) f (x) = ||x - 1| - 1|$