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.