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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q 62. (page 1006)

In the following exercise, find the inverse of each function.
$f (x) = \sqrt[5]{- 4 x - 3}$

Short Answer

Expert verified

The inverse of the given function is $f^{- 1} (x) = \frac{- x^{5} - 3}{4}$

Step by step solution

01

Step 1. Given information

The given function is $\sqrt[5]{- 4 x - 3}$

02

Step 2. Interchange the variable x and y.

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

$y = \sqrt[5]{- 4 x - 3}$

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

role="math" localid="1644936553145" $x = \sqrt[5]{- 4 y - 3}$

03

Step 3. solve for y

Solving for $y$ , we get:

role="math" localid="1644938527446" $x = \sqrt[5]{- 4 y - 3} x^{5} = - 4 y - 3 - 4 y = x^{5} + 3 y = \frac{- x^{5} - 3}{4}$

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

$f^{- 1} (x) = \frac{- x^{5} - 3}{4}$

04

Step 4. Verify that the functions are inverses.

Substituting $f (x)$ in role="math" localid="1644938933535" $f^{- 1} (f (x)) = x$ , we get:

$f^{- 1} (f (x)) = x f^{- 1} (\sqrt[5]{- 4 x - 3}) = x \sqrt[5]{- 4 (\frac{- x^{5} - 3}{4}) - 3} = x \sqrt[5]{x^{5} - 3 + 3} = x \sqrt[5]{x^{5}} = x x = x$

This is true.

Substituting $f^{- 1} (x)$ in role="math" localid="1644940940489" $f (f^{- 1} (x)) = x$ , we get:

$f (f^{- 1} (x)) = x f (\frac{- x^{5} - 3}{4}) = x \sqrt[5]{- 4 路 (\frac{- x^{5} - 3}{4}) - 3} \sqrt[5]{x^{5} + 3 - 3} = x \sqrt[5]{x^{5}} = x x = x$

This is also true.

Thus, the obtained inverse is correct.

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