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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 67 (page 269)

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{3 x + 4}{2 x - 3}$

Short Answer

Expert verified

The inverse off is $\frac{4 + 3 x}{2 x - 3} .$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = \frac{3 x + 4}{2 x - 3}$

We have to find its inverse and check the answer.

02

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

So, $y = \frac{3 x + 4}{2 x - 3}$

Interchange the variables

$x = \frac{3 y + 4}{2 y - 3}$

03

Step 3. Solve for y

$x = \frac{3 y + 4}{2 y - 3} 2 x y - 3 x = 3 y + 4 2 x y - 3 y = 4 + 3 x y (2 x - 3) = 4 + 3 x y = \frac{4 + 3 x}{2 x - 3}$

The inverse is $f^{- 1} (x) = \frac{4 + 3 x}{2 x - 3} .$

04

Step 4. Check the answer

We can check the answers by $f^{- 1} (f (x)) a n d f (f^{- 1} (x)) .$

So,

$f^{- 1} (f (x)) = f^{- 1} (\frac{3 x + 4}{2 x - 3}) = \frac{4 + 3 (\frac{3 x + 4}{2 x - 3})}{2 (\frac{3 x + 4}{2 x - 3}) - 3} = \frac{17 x}{17} = x$

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

Now,

$f (f^{- 1} (x)) = f (\frac{4 + 3 x}{2 x - 3}) = \frac{3 (\frac{4 + 3 x}{2 x - 3}) + 4}{2 (\frac{4 + 3 x}{2 x - 3}) - 3} = \frac{17 x}{17} = x$

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

Hence our answer is right.

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