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The millage rate is the amount of property tax per $1000 of the taxable value of a home. For a certain county the millage rate is 29 mil. A city within the county also imposes a flat fee of $101 per home. a. Write a function representing the total amount of property tax T(x) for a home with a taxable value of x thousand dollars. b. Write an equation for T-1(x). c. What does the inverse function represent in the context of this problem

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MrRoyal

Answer:

(a) [tex]T(x) = 101 + 0.029 x[/tex]

(b) [tex]T^{-1}(x) = \frac{1000(x - 101)}{29}[/tex]

(c) See explanation

Step-by-step explanation:

[tex]T(x) = 172 + \frac{29}{1000}x[/tex]

Given

[tex]Flat = 101[/tex]

[tex]Rate = 29\ per\ 1000[/tex]

Solving (a): The function for total amount

This is calculated as:

[tex]T(x) = Flat + Rate * x[/tex]

Where:

[tex]x \to[/tex] taxable value

So, we have:

[tex]T(x) = 101 + \frac{29}{1000} * x[/tex]

[tex]T(x) = 101 + 0.029 * x[/tex]

[tex]T(x) = 101 + 0.029 x[/tex]

Solving (b): The inverse function

[tex]T(x) = 101 + 0.029 x[/tex]

Rewrite as:

[tex]y = 101 + 0.029x[/tex]

Swap the variables

[tex]x = 101 + 0.029y[/tex]

Make y the subject

[tex]0.029y = x - 101[/tex]

Divide by 0.029

[tex]y = \frac{x - 101}{0.029}[/tex]

Multiply  by 1000/1000

[tex]y = \frac{1000(x - 101)}{29}[/tex]

Replace y with the inverse function

[tex]T^{-1}(x) = \frac{1000(x - 101)}{29}[/tex]

Solving (c): Interpret (b)

The inverse function, in this case; is the taxable value calculated from the amount of property tax

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