The function

Upper T left parenthesis x right parenthesis equals 0.15 left parenthesis x minus 1500 right parenthesis plus 150T(x)=0.15(x−1500)+150

represents the tax bill T of a single person whose adjusted gross income is x dollars for income over

$ 1500$1500

but not over

$53,600​,

in 2022.

​(a) What is the domain of this linear​ function?

​(b) What is a single​ filer's tax bill if the adjusted gross income is

$13,000​?

​(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

$ 3840

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.

---

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.

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 over $1500 but not over $53,600, in 2022. (a) What is the domain of this linear function? (b) What is a single filer’s tax bill if the adjusted gross income is $13,000? (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 $3840?

Explore the linear tax function T(x) = 0.15(x - 1500) + 150 for single filers, with a domain between $1500 and $53,600 in 2022. Learn how to calculate tax bills for various incomes, graph the function, and solve for income given a tax bill. Suitable for high school students in Grades 9-11.

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