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Exercises Functions

Example 10 from Algebra and Trigonometry, 3.7 Inverse Functions

Finding the Inverse of a Function Using Reflection about the Identity Line

Given the graph of $f (x)$ in Figure 9, sketch a graph of $f^{- 1} (x) .$

Graph of f^(-1)(x).

Figure 9

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 one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of $(0, \infty)$ and range of $(- \infty, \infty),$ so the inverse will have a domain of $(- \infty, \infty)$ and range of $(0, \infty) .$

If we reflect this graph over the line $y = x,$ the point $f^{- 1}$ reflects to $(0, 1)$ and the point $(4, 2)$ reflects to $(2, 4) .$ Sketching the inverse on the same axes as the original graph gives Figure 10.

Figure 10 The function and its inverse, showing reflection about the identity line

Your turn

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

Draw graphs of the functions $f$ and $f^{- 1}$ from Example 8.

How did it go?

PreviousSolving to Find an Inverse with Radicals Next Determining If Menu Price Lists Are Functions

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 3.7, 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.