Q 42. Verify that the functions fx=x-5... [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 42. (page 269)

Verify that the functions $f (x) = \frac{x - 5}{2 x + 3}$ and $g (x) = \frac{3 x + 5}{1 - 2 x}$ are inverses of each other by showing that $f (g (x)) = x$ and $g (f (x)) = x$ . Give any values ofx that need to be excluded from the domain of f and the domain ofg.

Short Answer

Expert verified

Functions f and g are inverses of one another .Exclude $x = - \frac{3}{2}$ from domain of f and $x = \frac{1}{2}$ from domain ofg.

Step by step solution

Step 1. Given information.

Given functions $f (x) = \frac{x - 5}{2 x + 3}$ and $g (x) = \frac{3 x + 5}{1 - 2 x}$ .

Step 2. Verify that the functions f and g and are inverses of each other.

Note that functions f and g are inverses of each other if $f (g (x)) = x$ and $g (f (x)) = x$ .

Prove $f (g (x)) = x$ as follows.

$\begin{array}{rcl} f (g (x)) & = & f (\frac{3 x + 5}{1 - 2 x}) \\ = & \frac{(\frac{3 x + 5}{1 - 2 x}) - 5}{2 (\frac{3 x + 5}{1 - 2 x}) + 3} \\ = & \frac{3 x + 5 - 5 + 10 x}{6 x + 10 + 3 - 6 x} \\ = & \frac{13 x}{13} \\ = & x \end{array}$

Prove $g (f (x)) = x$ as follows.

$\begin{array}{rcl} g (f (x)) & = & g (\frac{x - 5}{2 x + 3}) \\ = & \frac{3 (\frac{x - 5}{2 x + 3}) + 5}{1 - 2 (\frac{x - 5}{2 x + 3})} \\ = & \frac{3 x - 15 + 10 x + 15}{2 x + 3 - 2 x + 10} \\ = & \frac{13 x}{13} \\ = & x \end{array}$

Therefore, f and g are inverses of each other.

Step 3. Find values of x that need to be excluded from domain of f and g.

Note that domain of f is $x 鈮� - \frac{3}{2}$ and that of gis $x 鈮� \frac{1}{2}$ .

It follows that f exists for all real numbers except for $- \frac{3}{2}$ and g exists for all real numbers except for $\frac{1}{2}$ .

Therefore, $x = - \frac{3}{2}$ and $x = \frac{1}{2}$ need to be excluded from domain of fand g respectively.

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91影视

Verify that the functions $f (x) = \frac{x - 5}{2 x + 3}$ and $g (x) = \frac{3 x + 5}{1 - 2 x}$ are inverses of each other by showing that $f (g (x)) = x$ and $g (f (x)) = x$ . Give any values ofx that need to be excluded from the domain of f and the domain ofg.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Verify that the functions f and g and are inverses of each other.

Step 3. Find values of x that need to be excluded from domain of f and g.

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91影视

Verify that the functions fx=x-52x+3and gx=3x+51-2xare inverses of each other by showing that fgx=x and gfx=x. Give any values ofx that need to be excluded from the domain of f and the domain ofg.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Verify that the functions f and g and are inverses of each other.

Step 3. Find values of x that need to be excluded from domain of f and g.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Verify that the functions $f (x) = \frac{x - 5}{2 x + 3}$ and $g (x) = \frac{3 x + 5}{1 - 2 x}$ are inverses of each other by showing that $f (g (x)) = x$ and $g (f (x)) = x$ . Give any values ofx that need to be excluded from the domain of f and the domain ofg.