Problem of the Week #16

Problem 16: 1998-1999

Given

    (a)    Determine whether or not f(x) has an inverse function
    (b)    Discuss whether or not f(x) is monotonic using derivative(s)
    (c)    Find f^-1(x)
    (d)    Find (f^-1)'(12) Difficulty rating:

Solution: (a) The function clearly has an inverse because the graph passes the horizontal line test (and is therefore one-to-one).

(b) Clearly, the derivative of the function is x² + 5/3. Wherever this derivative is zero or undefined will be a critical number. Recall that these critical numbers form the points on the wiggle graph. Unfortunately, (or fortunately, depending on your viewpoint!) there is no place where f '(x) is undefined or equal to zero. (The graph of f '(x) never touches the x-axis.) Thus, there are no critical numbers, and the graph must always go in the same direction. Plug in an x-value into the derivative to determine this direction. I'll pick x = 1. Note that f '(1)=1+5/3, which is positive, so the graph is always increasing, but you could tell that when you graphed it in part (a), couldn't you?

(c) If you rewrite the function with a "y" instead of "f(x)", then switch x and y and try to solve for y (our technique for finding inverse functions), you'll find that it's quite impossible -- there are too many y's to isolate one on the left side of the equal sign. Therefore, using our methods, we cannot find the inverse function! How then are we to answer part (d)? How can we find the derivative of the inverse function if we can't find the stinkin' inverse function? There are other ways ....

(d) Recall from our discussions that

So, plug in x = 12. How are we to find f^-1(12), though? Think about it--if we can plug 12 into the inverse of f, we must be able to get 12 as an answer from f. In other words, if 12 is an input for the inverse, it was an output for the function. Therefore, I can say

Wherever the equation on the right crosses the x-axis is the answer we seek. That x-value will produce a y-value of 12 in the function! Therefore (if we reverse the coordinates) if x = 12, the result will be the y for the inverse function. Use your calculator to find the "zero" or "root", utilizing the 2nd-Calc menu option #2 on your TI-82 or TI-83 graphing calculator. We see that x = 2.5771. Therefore, (2.5771, 12) is a point on f, and (12, 2.5771) is a point on f^-1. Finally, this given us the information we needed, as f^-1(12)=2.5771.

The last step in this problem is finding 1/(f '(2.5771)). We plug 2.5771 into f ' to get 8.308111, Lastly, 1/8.308111 is .1204. Therefore, (f^-1)'(12)=.1204.

Superbowl of Calculus