To address the question, let's tackle each part systematically.
### Part (a):
The question asks us to interpret [tex]$D^{-1}(x)$[/tex]. First, let's understand the meaning of [tex]$D^{-1}(x)$[/tex].
By definition, [tex]$D(t)$[/tex] gives the distance traveled from Glen City after [tex]\( t \)[/tex] hours of walking. Therefore, the inverse function [tex]$D^{-1}(x)$[/tex] tells us the number of hours it took to be [tex]\( x \)[/tex] kilometers from Glen City.
Thus, the correct statement describing [tex]$D^{-1}(x)$[/tex] is:
- The amount of time she has walked (in hours) when she is [tex]\( x \)[/tex] kilometers from Glen City.
### Part (b):
We are given that the inverse function is:
[tex]\[ D^{-1}(x) = 3.1 - 0.2x \][/tex]
### Part (c):
To find [tex]$D^{-1}(8.5)$[/tex], we substitute [tex]\( x = 8.5 \)[/tex] into the given inverse function:
[tex]\[ D^{-1}(8.5) = 3.1 - 0.2 \times 8.5 \][/tex]
After performing the substitution and simplification (as calculated before):
[tex]\[ D^{-1}(8.5) = 3.1 - 1.7 = 1.4 \][/tex]
Therefore, the result is:
[tex]\[ D^{-1}(8.5) = 1.4 \][/tex]
So, the complete solution is:
- (a) The amount of time she has walked (in hours) when she is [tex]\( x \)[/tex] kilometers from Glen City.
- (b) [tex]\( D^{-1}(x) = 3.1 - 0.2x \)[/tex]
- (c) [tex]\( D^{-1}(8.5) = 1.4 \)[/tex]
That's the detailed step-by-step solution to the problem.
