Let's analyze each of the options provided to determine which one best explains the meaning of the inverse function [tex]\( f^{-1} \)[/tex].
Given that [tex]\( N = f(t) \)[/tex] represents the total number of inches of snow that fall in the first [tex]\( t \)[/tex] days of January, [tex]\( f(t) \)[/tex] is a function that takes the number of days [tex]\( t \)[/tex] as input and produces the total snowfall [tex]\( N \)[/tex] as output.
The inverse function [tex]\( f^{-1} \)[/tex] reverses this relationship. It takes the total snowfall [tex]\( N \)[/tex] as input and produces the number of days [tex]\( t \)[/tex] as output. Let's examine each option:
A. The number of inches of snow on the ground after [tex]\( t \)[/tex] days:
- This statement is describing the original function [tex]\( f(t) \)[/tex], not the inverse function [tex]\( f^{-1}(N) \)[/tex].
B. The days for which there are [tex]\( N \)[/tex] inches of snow on the ground:
- This statement is somewhat unclear and imprecise as it could be interpreted in multiple ways.
C. The number of inches of snow accumulated in [tex]\( t \)[/tex] days:
- Similar to option A, this statement describes the original function [tex]\( f(t) \)[/tex].
D. The number of days it takes to accumulate [tex]\( N \)[/tex] inches of snow:
- This statement accurately describes the inverse function [tex]\( f^{-1}(N) \)[/tex]. The inverse function takes [tex]\( N \)[/tex] inches of snow as input and returns [tex]\( t \)[/tex], the number of days it takes to accumulate that amount of snow.
E. None of the above:
- Since option D accurately describes the inverse function, we can conclude that option E is not correct.
Thus, the best explanation of the meaning of the inverse function [tex]\( f^{-1} \)[/tex] is given by:
D. The number of days it takes to accumulate [tex]\( N \)[/tex] inches of snow.
