Q. 43 In Problems 29 鈥� 44, for the g... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 43 (page 257)

In Problems 29 鈥� 44, for the given functions f and g, find: (a) $f 鈭� g$ (b) $g 鈭� f$ (c) $f 鈭� f$
(d) $g 鈭� g$
State the domain of each composite function.
$f (x) = \frac{x - 5}{x + 1}, g (x) = \frac{x + 2}{x - 3}$

Short Answer

Expert verified

(a) $f 鈭� g = \frac{- 4 x - 13}{- 2 x + 11}$ and domain is $R - \{\frac{1}{2}\}$

(b) $g 鈭� f = \frac{3 x - 3}{- 2 x - 8}$ and domain is $R - \{- 4\}$

(d) $g 鈭� g = \frac{3 x - 4}{- 2 x + 11}$ and domain is $R - \{\frac{11}{2}\}$

Step by step solution

Step 1. Given Information

The given functions are $f (x) = \frac{x - 5}{x + 1}, g (x) = \frac{x + 2}{x - 3}$

Part (a) Step 1. Finding f∘g

$f 鈭� g = f (g (x)) = f (\frac{x + 2}{x - 3}) f 鈭� g = \frac{\frac{x + 2}{x - 3} - 5}{\frac{x + 2}{x - 3} - 1} f 鈭� g = \frac{\frac{x + 2 - 5 x + 15}{x - 3}}{\frac{x + 2 + x - 3}{x - 3}} f 鈭� g = \frac{- 4 x + 17}{2 x - 1}$

Part (a) Step 2. Finding domain of f∘g

Put $2 x - 1 = 0 2 x = 1 x = \frac{1}{2}$

The domain is the set of all real numbers except $\frac{1}{2} .$

Part (b) Step 1. Finding g∘f

$g 鈭� f = g (f (x)) = g (\frac{x - 5}{x + 1}) g 鈭� f = \frac{\frac{x - 5}{x + 1} + 2}{\frac{x - 5}{x + 1} - 3} g 鈭� f = \frac{\frac{x - 5 + 2 x + 2}{x + 1}}{\frac{x - 5 - 3 x - 3}{x + 1}} g 鈭� f = \frac{3 x - 3}{- 2 x - 8}$

Part (b) Step 2. Finding domain of g∘f

Put $- 2 x - 8 = 0 - 2 x = 8 x = - 4$

The domain is the set of all real numbers except $- 2 .$

Part (c) Step 1. Finding f∘f

$f 鈭� f = f (f (x)) = f (\frac{x - 5}{x + 1}) f 鈭� f = \frac{\frac{x - 5}{x + 1} - 5}{\frac{x - 5}{x + 1} + 1} = \frac{x - 5 - 5 x - 5}{x - 5 + x + 1} f 鈭� f = \frac{- 4 x - 10}{2 x - 4} = \frac{- 2 (x + 5)}{2 (x - 2)} = \frac{- x - 5}{x - 2}$

Part (c) Step 2. Finding domain

Put $x - 2 = 0 x = 2$

The domain is the set of all real number except $2 .$

Part (d) Step 2. Finding g∘g

$g 鈭� g = g (g (x)) = g (\frac{x + 2}{x - 3}) g 鈭� g = \frac{\frac{x + 2}{x - 3} + 2}{\frac{x + 2}{x - 3} - 3} = \frac{x + 2 + 2 x - 6}{x + 2 - 3 x + 9} g 鈭� g = \frac{3 x - 4}{- 2 x + 11}$

Part (d) Step 2. Finding domain of g∘g

Put $- 2 x + 11 = 0 - 2 x = - 11 x = \frac{11}{2}$

The domain is the set of all real numbers except $\frac{11}{2} .$

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91影视

In Problems 29 鈥� 44, for the given functions f and g, find: (a) $f 鈭� g$ (b) $g 鈭� f$ (c) $f 鈭� f$
(d) $g 鈭� g$
State the domain of each composite function.
$f (x) = \frac{x - 5}{x + 1}, g (x) = \frac{x + 2}{x - 3}$

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. Finding f∘g

Part (a) Step 2. Finding domain of f∘g

Part (b) Step 1. Finding g∘f

Part (b) Step 2. Finding domain of g∘f

Part (c) Step 1. Finding f∘f

Part (c) Step 2. Finding domain

Part (d) Step 2. Finding g∘g

Part (d) Step 2. Finding domain of g∘g

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91影视

In Problems 29 鈥� 44, for the given functions f and g, find: (a) f鈭�g(b) g鈭�f(c) f鈭�f(d) g鈭�gState the domain of each composite function.fx=x-5x+1,gx=x+2x-3

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. Finding f∘g

Part (a) Step 2. Finding domain of f∘g

Part (b) Step 1. Finding g∘f

Part (b) Step 2. Finding domain of g∘f

Part (c) Step 1. Finding f∘f

Part (c) Step 2. Finding domain

Part (d) Step 2. Finding g∘g

Part (d) Step 2. Finding domain of g∘g

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Calculus

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Problems 29 鈥� 44, for the given functions f and g, find: (a) $f 鈭� g$ (b) $g 鈭� f$ (c) $f 鈭� f$
(d) $g 鈭� g$
State the domain of each composite function.
$f (x) = \frac{x - 5}{x + 1}, g (x) = \frac{x + 2}{x - 3}$