Chapter 5: Q. 61 (page 269)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

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

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

Interchange the variables

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

Step 3. Solve for y

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

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

Step 4. Check the answer

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

So,

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

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

Now,

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

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

Therefore, our answer is right.

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

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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In Problems 61–72, the function f is one-to-one. Find its inverse and check your answer.f(x)=23+x

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

Statistics

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Theoretical and Mathematical Physics

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

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