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Let's solve each part of the problem step by step based on the function $ T(x) = 0.15(x - 1500) + 150 $, which represents the tax bill $ T $ of a single person with an adjusted gross income $ x $ between $1,500 and $64,400 inclusive.

### (a) Domain of the linear function
The domain of the function is the range of values that $ x $, the adjusted gross income, can take. The income range is given as between $1,500 and $64,400. Hence, the domain is:

\[
[1500, 64400]
\]

### (b) Tax bill if adjusted gross income is $16,000
We need to substitute $ x = 16000 $ into the function $ T(x) $:

\[
T(16000) = 0.15(16000 - 1500) + 150
\]
\[
T(16000) = 0.15(14500) + 150 = 2175 + 150 = 2325
\]

So, the tax bill would be **$2,325**.

### (c) Which variable is independent and which is dependent?
- The adjusted gross income $ x $ is the **independent variable**.
- The tax bill $ T(x) $ is the **dependent variable** since it depends on the income.

Thus, **option A** is the correct answer.

### (d) Graph the function
We know the function $ T(x) = 0.15(x - 1500) + 150 $ is linear, starting at $ x = 1500 $ and going up to $ x = 64400 $. From the options provided:
- The function starts at a low value (when $ x = 1500 $) and increases as $ x $ increases.
- Since it's a linear function with a positive slope, **option C** is the correct graph.

### (e) Adjusted gross income when the tax bill is $4,350
We need to solve for $ x $ when $ T(x) = 4350 $:

\[
4350 = 0.15(x - 1500) + 150
\]
\[
4350 - 150 = 0.15(x - 1500)
\]
\[
4200 = 0.15(x - 1500)
\]
\[
x - 1500 = \frac{4200}{0.15} = 28000
\]
\[
x = 28000 + 1500 = 29500
\]

So, the adjusted gross income would be **$29,500**.

---

Would you like more details or have any questions about any part of the solution? Here are 5 related questions to deepen your understanding:

1. What does the slope of the linear function represent in this context?
2. How would the tax bill change if the income limit was increased to $70,000?
3. Can you calculate the tax bill for an income of $40,000?
4. How would the function change if the tax rate (0.15) were increased to 0.20?
5. What would be the adjusted gross income if the tax bill were $5,000?

**Tip:** For linear functions, the coefficient of $ x $ (the slope) directly shows how much the dependent variable changes with each unit increase in the independent variable.

The function T(x) = 0.15(x − 1500) + 150 represents the tax bill T of a single person whose adjusted gross income is x dollars for income between $1,500 and $64,400. Inclusive.
(a) What is the domain of this linear function?
(b) What is a single filer’s tax bill if the adjusted gross income is $16,000?
(c) Which variable is independent and which is dependent?
(d) Graph the linear function over the domain specified in part (a).
(e) What is a single filer’s adjusted gross income if the tax bill is $4,350?

This problem explores the linear function T(x) = 0.15(x - 1500) + 150, which models the tax bill of a single filer based on adjusted gross income between $1,500 and $64,400. Learn how to calculate the tax bill, determine the function's domain, and graph the equation. Suitable for students in Grades 9-11.

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