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

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

Step by step solution

Part (a) Step 1. Given information is:

$f (x) = | x |$

Part (a) Step 2. Piecewise Defined Function

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

Part (b) Step 1. Given information is:

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

Part (b) Step 2. Piecewise Defined Function

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

Part (c) Step 1. Given information is:

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

Part (c) Step 2. Piecewise Defined Function

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

Part (d) Step 1. Given information is:

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

Part (d) Step 2. Piecewise Defined Function

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

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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

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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

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Pure Maths

Statistics

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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