Let's break down and solve the given problem step by step. The function for the tax bill $T(x)$ is:

$T(x) = 0.15(x - 1500) + 150$

This represents the tax bill $T(x)$ for a single filer with adjusted gross income $x$ dollars, for incomes above $1500 but not exceeding $53,600 in 2022.

(a) Domain of the linear function

The domain specifies the range of values for the independent variable $x$ (the adjusted gross income). According to the problem, this function applies to incomes over $1500 but not exceeding $53,600.

Therefore, the domain of the function is: $\text{Domain} = [1500, 53,600]$

(b) Single filer’s tax bill if the adjusted gross income is $13,000

To find the tax bill when the adjusted gross income is $x = 13,000$, we substitute this value into the function:

$T(13,000) = 0.15(13,000 - 1500) + 150$ $T(13,000) = 0.15(11,500) + 150$ $T(13,000) = 1,725 + 150 = 1,875$

So, the tax bill for an income of $13,000 is $1,875.

(c) Independent and dependent variables

• Independent variable: The adjusted gross income $x$ is the independent variable, because it determines the tax bill.
• Dependent variable: The tax bill $T(x)$ is the dependent variable, as it depends on the value of $x$.

(d) Graphing the linear function

The function $T(x) = 0.15(x - 1500) + 150$ is a linear equation. To graph it:

• The domain is $[1500, 53,600]$.
• The slope of the line is $0.15$, and the y-intercept can be found by evaluating $T(x)$ when $x = 1500$: $T(1500) = 0.15(1500 - 1500) + 150 = 150$
• The graph will start at $(1500, 150)$ and rise with a slope of $0.15$ up to $x = 53,600$.

(e) Adjusted gross income if the tax bill is $3,840

To find the adjusted gross income when the tax bill is $T(x) = 3,840$, we solve for $x$:

$3,840 = 0.15(x - 1500) + 150$ Subtract 150 from both sides: $3,690 = 0.15(x - 1500)$ Divide by 0.15: $24,600 = x - 1500$ Add 1500 to both sides: $x = 26,100$

So, the adjusted gross income is $26,100 when the tax bill is $3,840.

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Do you want more details on any part of the solution or have any specific questions?

5 Related Questions:

1. How would the graph of the tax function change if the tax rate changed to 0.2 instead of 0.15?
2. What is the slope-intercept form of the equation $T(x) = 0.15(x - 1500) + 150$?
3. How would you find the inverse function of $T(x)$, relating income to tax?
4. What does the slope represent in the context of this tax function?
5. How does the y-intercept of 150 relate to the income levels below $1500?

Tip:

When dealing with linear functions, always check for the intercepts and slope to understand how the function behaves visually and contextually.