Q 41. Verify that the functions fx=2x+... [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 41. (page 269)

Verify that the functions $f (x) = \frac{2 x + 3}{x + 4}$ and $g (x) = \frac{4 x - 3}{2 - x}$ 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 $x = - 4$ from domain of f and $x = 2$ from domain of g.

Step by step solution

Step 1. Given information.

Given functions $f (x) = \frac{2 x + 3}{x + 4}$ and $g (x) = \frac{4 x - 3}{2 - x}$ .

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.

$\begin{array}{rcl} f (g (x)) & = & f (\frac{4 x - 3}{2 - x}) \\ = & \frac{2 (\frac{4 x - 3}{2 - x}) + 3}{(\frac{4 x - 3}{2 - x}) + 4} \\ = & \frac{8 x - 6 + 6 - 3 x}{4 x - 3 + 8 - 4 x} \\ = & \frac{5 x}{5} \\ = & x \end{array}$

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

$\begin{array}{rcl} g (f (x)) & = & g (\frac{2 x + 3}{x + 4}) \\ = & \frac{4 (\frac{2 x + 3}{x + 4}) - 3}{2 - (\frac{2 x + 3}{x + 4})} \\ = & \frac{8 x + 12 - 3 x - 12}{2 x + 8 - 2 x - 3} \\ = & \frac{5 x}{5} \\ = & x \end{array}$

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 $x 鈮� - 4$ and that of gis $x 鈮� 2$ .

It follows that f exists for all real numbers except for $- 4$ and g exists for all real numbers except for 2.

Therefore, $x = - 4$ and $x = 2$ need to be excluded from domain of fand g respectively.

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91影视

Verify that the functions $f (x) = \frac{2 x + 3}{x + 4}$ and $g (x) = \frac{4 x - 3}{2 - x}$ 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.

One App. One Place for Learning.

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91影视

Verify that the functions fx=2x+3x+4and gx=4x-32-xare 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

Pure Maths

Probability and Statistics

Discrete Mathematics

Applied Mathematics

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

Verify that the functions $f (x) = \frac{2 x + 3}{x + 4}$ and $g (x) = \frac{4 x - 3}{2 - x}$ 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.