Q. 69 In Problems 61鈥�72, the functio... [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. 69 (page 269)

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{2 x + 3}{x + 2}$

Short Answer

Expert verified

The inverse of fis $\frac{3 - 2 x}{x - 2} .$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \frac{2 x + 3}{x + 2}$

We have to find its inverse and check the answer.

Step 2. Replace f(x) with y and interchange the variables x and y

So, $y = \frac{2 x + 3}{x + 2}$

Interchange the variables

$x = \frac{2 y + 3}{y + 2}$

Step 3. Solve for y

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

The inverse isrole="math" localid="1646213189839" $f^{- 1} (x) = \frac{2 x - 3}{2 - x}$

Step 4. Check the answer

We check the answer by $f^{- 1} (f (x))$ and $f (f^{- 1} (x))$ .

So,

$f^{- 1} (f (x)) = f^{- 1} (\frac{2 x + 3}{x + 2}) = \frac{3 - 2 (\frac{2 x + 3}{x + 2})}{(\frac{2 x + 3}{x + 2}) - 2} = \frac{3 - (\frac{4 x + 6}{x + 2})}{\frac{2 x + 3 - 2 x - 4}{x + 2}} = \frac{\frac{3 x + 6 - 4 x - 6}{x + 2}}{\frac{2 x + 3 - 2 x - 4}{x + 2}} = \frac{- x}{- 1} = x$

Thus, $f^{- 1} (f (x)) = x$

Now,

$f (f^{- 1} (x)) = f (\frac{3 - 2 x}{x - 2}) = \frac{2 (\frac{3 - 2 x}{x - 2}) + 3}{(\frac{3 - 2 x}{x - 2}) + 2} = \frac{(\frac{6 - 4 x}{x - 2}) + 3}{\frac{3 - 2 x + 2 x - 4}{x - 2}} = \frac{\frac{6 - 4 x + 3 x - 6}{x - 2}}{\frac{- 1}{x - 2}} = \frac{- x}{- 1} = x$

Thus, $f (f^{- 1} (x)) = x$

Therefore, our answer is right.

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91影视

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{2 x + 3}{x + 2}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Replace f(x) with y and interchange the variables x and y

Step 3. Solve for y

Step 4. Check the answer

One App. One Place for Learning.

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91影视

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.f(x)=2x+3x+2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Replace f(x) with y and interchange the variables x and y

Step 3. Solve for y

Step 4. Check the answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Statistics

Pure Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{2 x + 3}{x + 2}$