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Exercises Polynomials and factoring

Example 5 from Algebra and Trigonometry, 5.7 Inverses and Radical Functions

Finding the Inverse of a Radical Function

Restrict the domain of the function $f (x) = \sqrt{x - 4}$ and then find the inverse.

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
Note that the original function has range $f (x) \geq 0.$ Replace $f (x)$ with $y,$ then solve for $x .$

$\begin{matrix} y & = & \sqrt{x - 4} & Replace f (x) with y . \\ x & = & \sqrt{y - 4} & Interchange x and y . \\ x & = & \sqrt{y - 4} & Square each side . \\ x^{2} & = & y - 4 & Add 4 . \\ x^{2} + 4 & = & y & Rename the function f^{- 1} (x) . \\ f^{- 1} (x) & = & x^{2} + 4 \end{matrix}$
Recall that the domain of this function must be limited to the range of the original function.

$f^{- 1} (x) = x^{2} + 4, x \geq 0$

Your turn

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

Restrict the domain and then find the inverse of the function $f (x) = \sqrt{2 x + 3} .$

How did it go?

PreviousFinding the Inverse of a Quadratic Function When the Restriction Is Not Specified Next Solving an Application with a Cubic 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.