Chapter 10: Q. 54 (page 1005)

In the following exercise, find the inverse of each function.
$f (x) = \frac{1}{x - 6}$

Short Answer

Expert verified

The inverse of the given function is $f^{- 1} (x) = \frac{1}{x} + 6$

Step by step solution

Step 1. Given information

The given function is $f (x) = \frac{1}{x - 6}$

Step 2. Interchange the variable y and x

Substituting $f (x) = y$ , we get:

$y = \frac{1}{x - 6}$

interchanging the variable $x$ and $y$ , we get:

$x = \frac{1}{y - 6}$

Step 3. Solve for y

Solving for $y$ , we get:

$x = \frac{1}{y - 6} y - 6 = \frac{1}{x} y = \frac{1}{x} + 6$

Substituting $y = f^{- 1} (x)$ , we get:

role="math" localid="1644903804187" $f^{- 1} (x) = \frac{1}{x} + 6$

Step 4. Verify that the functions are inverses.

Substituting $f (x)$ in $f^{- 1} (f (x)) = x$ , we get:

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

This is true.

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

This is True.

The obtained inverse is correct.

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In the following exercise, find the inverse of each function.
$f (x) = \frac{1}{x - 6}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Interchange the variable y and x

Step 3. Solve for y

Step 4. Verify that the functions are inverses.

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In the following exercise, find the inverse of each function. fx=1x-6

Short Answer

Step by step solution

Step 1. Given information

Step 2. Interchange the variable y and x

Step 3. Solve for y

Step 4. Verify that the functions are inverses.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

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Study anywhere. Anytime. Across all devices.

In the following exercise, find the inverse of each function.
$f (x) = \frac{1}{x - 6}$