Q. 44 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. 44 (page 257)

In Problems 29 鈥� 44, for the given functions fand g, find: $(a) f 鈭� g (b) g 鈭� f (c) f 鈭� f (d) g 鈭� g$
localid="1647279857360" $f (x) = \frac{2 x - 1}{x - 2}, g (x) = \frac{x + 4}{2 x - 5}$

Short Answer

Expert verified

(a) $f 鈭� g = \frac{13}{- 3 x + 4}$ and domain is $R - \{\frac{4}{3}\}$

(b) $g 鈭� f = \frac{6 x - 9}{- x + 8}$ and domain is $R - \{8\}$

(d) $g 鈭� g = \frac{9 x - 16}{- 8 x + 33}$ and domain is $R - \{\frac{33}{8}\}$

Step by step solution

Step 1. Given Information

The given functions are $f (x) = \frac{2 x - 1}{x - 2}, g (x) = \frac{x + 4}{2 x - 5}$

Part (a) Step 1. Finding f∘g

$f 鈭� g = f (g (x)) = f (\frac{x + 4}{2 x - 5}) f 鈭� g = \frac{2 (\frac{x + 4}{2 x - 5}) - 1}{\frac{x + 4}{2 x - 5} - 2} f 鈭� g = \frac{\frac{2 x + 8 - 2 x + 5}{2 x - 5}}{\frac{x + 4 - 4 x + 10}{2 x - 5}} f 鈭� g = \frac{13}{- 3 x + 14}$

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

Put $- 3 x + 14 = 0 - 3 x = - 14 x = \frac{14}{3}$

The domain is the set of all real number except $\frac{14}{3} .$

Part (b) Step 1. Finding g∘f

$g 鈭� f = g (f (x)) = g (\frac{2 x - 1}{x - 2}) g 鈭� f = \frac{\frac{2 x - 1}{x - 2} + 4}{2 (\frac{2 x - 1}{x - 2}) - 5} g 鈭� f = \frac{\frac{2 x - 1 + 4 x - 8}{x - 2}}{\frac{4 x - 2 - 5 x + 10}{x - 2}} g 鈭� f = \frac{6 x - 9}{- x + 8}$

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

Put $- x + 8 = 0 - x = - 8 x = 8$

The domain is the set of all real number except

Part (c) Step 1. Finding f∘f

$f 鈭� f = f (f (x)) = f (\frac{2 x - 1}{x - 2}) f 鈭� f = \frac{2 . \frac{2 x - 1}{x - 2} - 1}{\frac{2 x - 1}{x - 2} - 2} f 鈭� f = \frac{4 x - 2 - x + 2}{2 x - 1 - 2 x + 4} = \frac{3 x}{3} = x$

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

We get $f 鈭� f = x$

This is an identity function. The domain of identity function is the set of Real number.

Part (d) Step 1. Finding g∘g

$g 鈭� g = g (g (x)) = g (\frac{x + 4}{2 x - 5}) g 鈭� g = \frac{\frac{x + 4}{2 x - 5} + 4}{2 (\frac{x + 4}{2 x - 5}) - 5} = \frac{\frac{x + 4 + 8 x - 20}{2 x - 5}}{\frac{2 x + 8 - 10 x + 25}{2 x - 5}} g 鈭� g = \frac{9 x - 16}{- 8 x + 33}$

Part (d) Step 2. Finding domain

Put $- 8 x + 33 = 0 - 8 x = - 33 x = \frac{33}{8}$

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91影视

In Problems 29 鈥� 44, for the given functions fand g, find: $(a) f 鈭� g (b) g 鈭� f (c) f 鈭� f (d) g 鈭� g$
localid="1647279857360" $f (x) = \frac{2 x - 1}{x - 2}, g (x) = \frac{x + 4}{2 x - 5}$

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 (d) Step 2. Finding domain of f∘f

Part (d) Step 1. Finding g∘g

Part (d) Step 2. Finding domain

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91影视

In Problems 29 鈥� 44, for the given functions fand g, find: (a)f鈭�g(b)g鈭�f(c)f鈭�f(d)g鈭�glocalid="1647279857360" fx=2x-1x-2,gx=x+42x-5

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 (d) Step 2. Finding domain of f∘f

Part (d) Step 1. Finding g∘g

Part (d) Step 2. Finding domain

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Pure Maths

Geometry

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Problems 29 鈥� 44, for the given functions fand g, find: $(a) f 鈭� g (b) g 鈭� f (c) f 鈭� f (d) g 鈭� g$
localid="1647279857360" $f (x) = \frac{2 x - 1}{x - 2}, g (x) = \frac{x + 4}{2 x - 5}$