Q.6 fx=3xx-2,gx=2x+1. Find(a) f鈭榞2... [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 12: Q.6 (page 840)

$f (x) = \frac{3 x}{x - 2}, g (x) = 2 x + 1$ . Find
(a) $(f 鈭� g) (2)$
(b) $(g 鈭� f) (2)$
(c) $(f 鈭� g) (x)$
(d) The domain of $(f 鈭� g) (x)$
(e) $(g 鈭� f) (x)$
(f) The domain of $(g 鈭� f) (x)$
(g) The function $g^{- 1}$ and its domain.
(h) The function $f^{- 1}$ and its domain.

Short Answer

Expert verified

(a) $(f 鈭� g) (2) = 5$

(b) $(g 鈭� f) (2) = 鈭�$

(d) The domain of $(f 鈭� g) (x)$ is $R - \{\frac{1}{2}\} .$

(e) $(g 鈭� f) (x) = \frac{7 x - 2}{x - 2}$

(f) The domain of $(g 鈭� f) (x)$ is $R - \{2\}$

(g) $g^{- 1} (x) = \frac{x - 1}{2}$ and domain is the set of all real numbers.

(h) $f^{- 1} (x) = \frac{2 x}{x - 3}$ and the domain $R - \{3\} .$

Step by step solution

Step 1. Given information

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

Part (a) Step 1. Finding f∘g2

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

Part (b) Step 1. Finding g∘f2

$(g 鈭� f) (2) = g (f (2)) = g (\frac{3 脳 2}{2 - 2}) = g (鈭�) = 鈭�$

Part (c) Step 1. f∘gx

$(f 鈭� g) (x) = f (g (x)) = f (2 x + 1) = \frac{3 (2 x + 1)}{2 x + 1 - 2} = \frac{6 x + 3}{2 x - 1}$

Part (d) Step 1. The domain of f∘gx

We get $(f 鈭� g) (x) = \frac{6 x + 3}{2 x - 1}$

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

The domain is $R - \{\frac{1}{2}\}$

Part (e) Step 1. Finding g∘fx

$(g 鈭� f) (x) = g (f (x)) = g (\frac{3 x}{x - 2}) = 2 (\frac{3 x}{x - 2}) + 1 = \frac{6 x}{x - 2} + 1 = \frac{6 x + x - 2}{x - 2} = \frac{7 x - 2}{x - 2}$

Part (f) Step 1. The domain of g∘fx

$(g 鈭� f) (x) = \frac{7 x - 2}{x - 2}$

Solve for $x - 2 = 0 x = 2$

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

Part (g) Step 1. Finding g-1

Put $y = g (x) y = 2 x + 1 2 x = y - 1 x = \frac{y - 1}{2}$

So, $g^{- 1} (x) = \frac{x - 1}{2}$

Part (g) Step 2. Finding domain of g-1

$g^{- 1} (x) = \frac{x - 1}{2}$

The denominator is a constant term. So the domain is the set of all real numbers.

Part (h) Step 1. Finding f-1

Put $y = f (x) y = \frac{3 x}{x - 2} 3 x = x y - 2 y 3 x - x y = - 2 y x (3 - y) = - 2 y x = \frac{2 y}{y - 3}$

So, $f^{- 1} (x) = \frac{2 x}{x - 3}$

Part (h) Step 2. Finding domain of f-1

We get $f^{- 1} (x) = \frac{2 x}{x - 3}$

Put $x - 3 = 0 x = 3$

The domain is the set of all real number except 3.

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Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Finding f∘g2

Part (b) Step 1. Finding g∘f2

Part (c) Step 1. f∘gx

Part (d) Step 1. The domain of f∘gx

Part (e) Step 1. Finding g∘fx

Part (f) Step 1. The domain of g∘fx

Part (g) Step 1. Finding g-1

Part (g) Step 2. Finding domain of g-1

Part (h) Step 1. Finding f-1

Part (h) Step 2. Finding domain of f-1

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fx=3xx-2,gx=2x+1. Find(a) f鈭�g2(b) g鈭�f2(c) f鈭�gx(d) The domain of f鈭�gx(e) g鈭�fx(f) The domain of g鈭�fx(g) The function g-1and its domain.(h) The function f-1and its domain.

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Finding f∘g2

Part (b) Step 1. Finding g∘f2

Part (c) Step 1. f∘gx

Part (d) Step 1. The domain of f∘gx

Part (e) Step 1. Finding g∘fx

Part (f) Step 1. The domain of g∘fx

Part (g) Step 1. Finding g-1

Part (g) Step 2. Finding domain of g-1

Part (h) Step 1. Finding f-1

Part (h) Step 2. Finding domain of f-1

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Geometry

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Applied Mathematics

Calculus

Probability and Statistics

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