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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 2: Q. 12 (page 124)

Find $f + g, f - g, f 鈰� g$ and $\frac{f}{g}$ for each pair of functions. State the domain of each of these functions.
role="math" localid="1645971066512" $f (x) = 2 - x g (x) = 3 x + 1$

Short Answer

Expert verified

For $f (x) = 2 - x g (x) = 3 x + 1$ , we get:

$f + g = 2 x + 3$ and its domain is the set of all real numbers .

$f - g = - 4 x + 1$ and its domain is the set of all real numbers .

$f 鈰� g = - 3 x^{2} + 5 x + 2$ and its domain is the set of all real numbers.

$\frac{f}{g} = \frac{2 - x}{3 x + 1}$ and ints domain is ${x | x 鈮� - \frac{1}{3}}$ .

Step by step solution

01

Step 1. Given information

We have been given:

$f (x) = 2 - x g (x) = 3 x + 1$

We have to find $f + g, f - g, f 鈰� g$ and $\frac{f}{g}$ for the pair of functions and state the domain of each of these functions.

02

Step 2. Find f+g and its domain.

We know $f + g = f (x) + g (x)$ .

Therefore,

$f + g = (2 - x) + (3 x + 1) = 2 - x + 3 x + 1 = 2 x + 3$

Its domain is the set of all real numbers.

03

Step 3. Find f-g and its domain.

We know $f - g = f (x) + g (x)$ .

$f - g = (2 - x) - (3 x + 1) = 2 - x - 3 x - 1 = 1 - 4 x$

Its domain is the set of all real numbers.

04

Step 4. Find f⋅g and its domain.

We know $f 鈰� g = f (x) 鈰� g (x)$ .

$f 鈰� g = (2 - x) (3 x + 1) = 2 (3 x + 1) - x (3 x + 1) = 6 x + 2 - 3 x^{2} - x = - 3 x^{2} + 5 x + 2$

Its domain is the set of all real numbers.

05

Step 5. Find fg and its domain.

We know $\frac{f}{g} = \frac{f (x)}{g (x)}$ .

Therefore,

role="math" localid="1645971854262" $\frac{f}{g} = \frac{2 - x}{3 x + 1}$

Its domain will be the set of all real numbers except those which make the denominator zero.

$3 x + 1 = 0 3 x = - 1 x = - \frac{1}{3}$

Domain is ${x | x 鈮� - \frac{1}{3}}$ .

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