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!