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Exercises Polynomials and factoring

Example 2 from Algebra and Trigonometry, 5.7 Inverses and Radical Functions

Finding the Inverse of a Cubic Function

Find the inverse of the function $f (x) = 5 x^{3} + 1.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
This is a transformation of the basic cubic toolkit function, and based on our knowledge of that function, we know it is one-to-one. Solving for the inverse by solving for $x .$

$\begin{matrix} y & = & 5 x^{3} + 1 \\ x & = & 5 y^{3} + 1 \\ x - 1 & = & 5 y^{3} \\ \frac{x - 1}{5} & = & y^{3} \\ f^{- 1} (x) & = & \sqrt[3]{\frac{x - 1}{5}} \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the inverse function of $f (x) = \sqrt[3]{x + 4} .$

How did it go?

PreviousVerifying Inverse Functions Next Restricting the Domain to Find the Inverse of a Polynomial Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.7, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.