Any help would be appreciated

Question

Let f(d) be the noise level (in decibels) of a rock concert at a distance d (in meters) from the concert speakers.

Explain in words the meaning of (f^-1)(120)=90 in the context of this problem.

Explain in words the meaning of (f-1)'(120) in the context of this problem. Would you expect (f^-1)'(120) to be positive or negative? Why?

Thanks!

Answer 1

Let "d" be the distance, and f(d) be "sl" or sound level. In the function f(d), an ordered pair (d, sl) represents a certain sound level given a distance. The inverse swaps these so would be (sl, d) or, the distance based on a certain sound level so f^-1(120) = 90 means a sound level of 120 dB would equate to a distance of 90 meters.

(f^-1)' is the rate of change of the distance with respect to the sound level so (f^-1)'(120) means how the distance varies as a function of the sound level when the sound level is 120 dB.

To help understand the last question, graph the sound level as a function of distance then flip it over the y=x line to get distance as a function of sound level:

You'll see that (f^-1)' is a negative (slopes are negative on the bottom graph). You can come up with the reason be thinking about slope as the change in y divided by the change in x