The problem involves linear depreciation, so let's break down each part.

Part 1: Finding the rule of the depreciation function $f(x)$

You are given:

• Initial value of the machine: $\$156,700$
• Scrap value after 11 years: $\$31,190$

Linear depreciation means the value of the machine decreases at a constant rate. To determine the depreciation rate:

1. Depreciation rate (slope) is the change in value over time: $\text{Depreciation rate} = \frac{\text{Initial value} - \text{Scrap value}}{\text{Time period}} = \frac{156700 - 31190}{11} = \frac{125510}{11} = 11410 \, \text{per year}$

2. Depreciation function: The machine's value $f(x)$ after $x$ years is given by: $f(x) = 156700 - 11410x$

So, the rule of the depreciation function is: $f(x) = 156700 - 11410x$

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Part 2: Domain of $f(x)$

The domain of the depreciation function corresponds to the time period over which the machine's value is being calculated. Since the machine depreciates over 11 years, the domain is: $\text{Domain of } f(x): \quad 0 \leq x \leq 11$ Where:

• $x = 0$ represents the time the machine is first purchased.
• $x = 11$ represents the time the machine reaches its scrap value.

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Part 3: Machine's worth after 7 years

To find the value of the machine after 7 years, substitute $x = 7$ into the depreciation function:

$f(7) = 156700 - 11410 \times 7 = 156700 - 79870 = 76830$

The machine will be worth $\$76,830$ after 7 years.

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Summary of Answers:

1. Depreciation function: $f(x) = 156700 - 11410x$
2. Domain: $0 \leq x \leq 11$
3. Worth after 7 years: $\$76,830$

Would you like more details on any of these calculations?

Here are 5 related questions:

1. How is the depreciation function derived?
2. How can depreciation be modeled in nonlinear situations?
3. What happens if the machine is sold before reaching its scrap value?
4. How is depreciation used in accounting?
5. How does salvage value affect depreciation?

Tip: The slope of a depreciation function is always negative because the value decreases over time.