Chapter 10: Q. 52 (page 1004)

In the following exercise, find the inverse of each function.
$f (x) = x^{3} + 6$

Short Answer

Expert verified

The inverse of the given function is $f^{- 1} (x) = \sqrt[3]{x - 6}$

Step by step solution

Step 1. Given information

The given function is $f (x) = x^{3} + 6$

Step 2. Interchange the variable x and y

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

$y = x^{3} + 6$

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

$x = y^{3} + 6$

Step 3. Solve for y

Solving for $y$ , we get:

$x = y^{3} + 6 y^{3} = x - 6 y = \sqrt[3]{(x - 6)}$

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

$f^{- 1} (x) = \sqrt[3]{x - 6}$

Step 4. verify that the functions are inverses.

Substituting $f (x)$ in localid="1644864594918" $f (f^{- 1} (x)) = x$ , we get:

$f^{- 1} (f (x)) = x f^{- 1} (x^{3} + 6)) = x \sqrt[3]{x^{3} + 6 - 6} = x \sqrt[3]{x^{3}} = x x = x$

This is True.

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

$f^{- 1} (x^{3} + 6) = x \sqrt[3]{x^{3} + 6 - 6} = x \sqrt[3]{x^{3}} = x x = x$

This is true.

Thus, the obtained inverse is correct

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In the following exercise, find the inverse of each function.
$f (x) = x^{3} + 6$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Interchange the variable x and y

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.f(x)=x3+6

Short Answer

Step by step solution

Step 1. Given information

Step 2. Interchange the variable x and y

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

Statistics

Pure Maths

Mechanics Maths

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In the following exercise, find the inverse of each function.
$f (x) = x^{3} + 6$