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Let's go through the problem step by step.

### Given Information:
The function provided is:

\[
T(x) = 0.15(x - 1500) + 150
\]

This represents the tax bill $ T $ of a single person whose adjusted gross income is $ x $ dollars, where $ x $ is between $1500 and $82,900 inclusive.

### Part (a): Domain of the function
The domain of a function is the set of all possible values of $ x $, which in this case represents the adjusted gross income. Since the problem specifies that $ x $ is between $1500 and $82,900, the domain is the interval:

\[
[1500, 82900]
\]

### Part (b): Tax bill for an adjusted gross income of $18,000
We substitute $ x = 18000 $ into the tax function $ T(x) $:

\[
T(18000) = 0.15(18000 - 1500) + 150
\]

Simplifying:

\[
T(18000) = 0.15(16500) + 150 = 2475 + 150 = 2625
\]

So, the tax bill for an income of $18,000 is **$2,625**.

### Part (c): Independent and dependent variables
- **Independent variable**: The adjusted gross income $ x $ (since it is what determines the tax bill).
- **Dependent variable**: The tax bill $ T(x) $ (since it depends on the value of $ x $).

### Part (d): Graph the linear function
The function is linear, meaning it is a straight line. To graph this function, plot the two points derived from the domain:
- When $ x = 1500 $, $ T(1500) = 0.15(1500 - 1500) + 150 = 150 $.
- When $ x = 82900 $, $ T(82900) = 0.15(82900 - 1500) + 150 = 12285 + 150 = 12435 $.

The graph will show a line between these points with a slope of 0.15.

### Part (e): Income if tax bill is $3,750
Set $ T(x) = 3750 $ and solve for $ x $:

\[
3750 = 0.15(x - 1500) + 150
\]

First, subtract 150 from both sides:

\[
3600 = 0.15(x - 1500)
\]

Now divide both sides by 0.15:

\[
x - 1500 = 24000
\]

Add 1500 to both sides:

\[
x = 25500
\]

So, the adjusted gross income is **$25,500** when the tax bill is $3,750.

---

Let me know if you need further details on any of the parts! Here are five follow-up questions:

1. How would the tax bill change if the tax rate were 20% instead of 15%?
2. What happens to the graph if the base tax (the constant 150) increases?
3. Can you calculate the income if the tax bill is $5,000?
4. How would the function change if there was no base tax (i.e., the +150 part was removed)?
5. What is the slope of the linear function, and what does it represent in this context?

**Tip**: To solve for unknowns in linear functions, isolate the variable step-by-step to keep things clear!

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 $1500 and $82,900. (a) What is the domain of this function? (b) What is a single filer's tax bill if the adjusted gross income is $18,000? (c) Which variable is independent and which is dependent? (d) Graph the function. (e) What is a single filer's income if the tax bill is $3750?

Learn how to solve a tax function problem where T(x) = 0.15(x - 1500) + 150 represents a tax bill based on adjusted gross income. This problem includes finding the domain, calculating the tax bill for a specific income, identifying independent and dependent variables, and solving for income given a tax bill.

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