Given the equation [tex]\( P = f(x) \)[/tex], where [tex]\( P \)[/tex] represents the profit in thousands of dollars and [tex]\( x \)[/tex] represents the number of units sold, we need to find the interpretation of [tex]\( f^{-1}(14) = 470 \)[/tex].
1. Let's break down the given equation [tex]\( f^{-1}(14) = 470 \)[/tex]:
- The function [tex]\( f^{-1} \)[/tex] is the inverse of [tex]\( f \)[/tex], meaning that it reverses the operation of [tex]\( f \)[/tex].
- When we say [tex]\( f^{-1}(14) = 470 \)[/tex], it implies that [tex]\( f(470) = 14 \)[/tex]. This tells us that when the number of units sold is 470, the profit is 14 (in thousands of dollars).
2. Convert the profit from thousands of dollars to actual dollars:
- Since the profit [tex]\( P \)[/tex] is given in thousands of dollars, [tex]\( 14 \)[/tex] (in thousands) translates to [tex]\( 14,000 \)[/tex] dollars.
Given these analyses, the statement that makes the most sense is:
- "If the company has a profit of \[tex]$ 14,000, they will have sold 470 units."
Thus, the correct interpretation is:
If the company has a profit of \$[/tex] 14,000, they will have sold 470 units.
