Chapter 10: Q. 57 (page 1005)

In the following exercise, find the inverse of each function.
$f (x) = \sqrt[3]{x - 3}$

Short Answer

Expert verified

The inverse of the given function is $f^{- 1} (x) = x^{3} + 3$

Step by step solution

Step 1. Given information

The given function is $f (x) = \sqrt[3]{x - 3}$

Step 2. Interchange the variable x and y

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

$y = \sqrt[3]{x - 3}$

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

$x = \sqrt[3]{y - 3}$

Step 3. Solve for y

Solving for $y$ , we get:

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

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

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

Step 4. Verify that the functions are inverses.

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

$f^{- 1} (f (x)) = x f^{- 1} (\sqrt[3]{x - 3}) = x {(\sqrt[3]{x - 3})}^{2} + 3 = x x - 3 + 3 = x x = x$

This is True

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

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

This is true.

Thus, the obtained inverse is correct

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In the following exercise, find the inverse of each function.
$f (x) = \sqrt[3]{x - 3}$

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.fx=x-33

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

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In the following exercise, find the inverse of each function.
$f (x) = \sqrt[3]{x - 3}$