Chapter 5: Q. 68 (page 269)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

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

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

Interchange the variables

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

Step 3. Solve for y

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

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

Step 4. Check the answer

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

So,

$f^{- 1} (f (x)) = f^{- 1} (\frac{2 x - 3}{x + 4}) = \frac{4 (\frac{2 x - 3}{x + 4}) + 3}{2 - (\frac{2 x - 3}{x + 4})} = \frac{11 x}{11} = x$

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

Now,

$f (f^{- 1} (x)) = f (\frac{4 x + 3}{2 - x}) = \frac{2 (\frac{4 x + 3}{2 - x}) - 3}{(\frac{4 x + 3}{2 - x}) + 4} = \frac{11 x}{11} = x$

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

Hence our result is right.

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

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

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Most popular questions from this chapter

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In Problems 61–72, the function f is one-to-one. Find its inverse and check your answer.f(x)=2x-3x+4

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

Pure Maths

Geometry

Decision Maths

Probability and Statistics

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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