Question

1. the number of parts produced by an automotive company can be modeled by a function, ( g ), where ( g(t) ) represents the number of parts produced (in hundreds), ( t ) hours into the workday. interpret the meaning of ( g^{-1}(7) = 3 ) in context.

Explanation:

Step1: Recall inverse function definition

The inverse function \( g^{-1}(y) = t \) means that if \( g(t) = y \), then \( g^{-1}(y) = t \). Here, \( g(t) \) is the number of parts (in hundreds) at time \( t \) hours.

Step2: Apply to \( g^{-1}(7) = 3 \)

For \( g^{-1}(7)=3 \), by the inverse function definition, this implies \( g(3)=7 \). Now, interpret \( g(3)=7 \): \( t = 3 \) hours into the workday, \( g(3) = 7 \) (hundreds of parts). So reversing it, \( g^{-1}(7)=3 \) means when the number of parts produced is 7 hundred (i.e., 700 parts), the time into the workday is 3 hours.

Answer:

When the number of parts produced is 700 (since \( g(t) \) is in hundreds, 7 hundreds = 700), it takes 3 hours into the workday to produce that many parts. In other words, 3 hours after the start of the workday, the automotive company has produced 700 parts (or 7 hundred parts).