16 A machine is now worth \( \$ 153,800 \) and will be depreciated linearly over a 9 -year period, at which time it will be worth \( \$ 41,930 \) as scrap. (a) Find the rule of depreciation function \( f \). (b) What is the domain of \( f \) ? (c) What will the machine be worth in 7 years? (a) Find the rule of depreciation function \( f \). \( f(x)=\square \) (Do not include the \( \$ \) symbol in your answer.) (b) What is the domain of \( f \) ? \( \square \) (Type your answer in interval notation.) (c) What will the machine be worth in 7 years? \( \$ \square \)

To solve the problem, let's start with the depreciation function. (a) The initial value of the machine is \( \$ 153,800 \) and its value at the end of 9 years will be \( \$ 41,930 \). The total depreciation over 9 years is: \[ 153,800 - 41,930 = 111,870 \] The yearly depreciation rate is: \[ \frac{111,870}{9} = 12,430 \] The rule of depreciation function \( f(x) \) can be defined as: \[ f(x) = 153,800 - 12,430x \] where \( x \) is the number of years since purchase. (b) The domain of \( f \) is defined over the time period of depreciation, which is from when the machine is first purchased (\( x = 0 \)) until the end of the 9 years (\( x = 9 \)). Therefore, the domain in interval notation is: \[ [0, 9] \] (c) To find out what the machine will be worth in 7 years, substitute \( x = 7 \) into the depreciation function: \[ f(7) = 153,800 - 12,430 \cdot 7 = 153,800 - 87,010 = 65,790 \] Thus, in 7 years, the machine will be worth \( \$ 65,790 \). To summarize your requested answers: (a) \( f(x) = 153800 - 12430x \) (b) \( [0, 9] \) (c) \( \$ 65,790 \) And remember, machines might depreciate, but your math skills sure won't when you keep practicing!