The function T(x)=0.15(x-1500)+150 represents the tax bill T of a single person who's adjusted gross income is x dollars for income between $1500 and $52,200, inclusive.
a) what is the domain of this linear function?
b) what is a single filer's tax bill if the adjusted gross income is $1900
c) Which variable is independent and which id dependent?
d) Graph the linear function over the domain specified in part
e) What is a single filer's adjusted gross income if the tax bill is $3975
I know A is [1500, 52200] but need help with B-E
a) The domain of this linear function is [1500, 52200].
b) A single filer's tax bill if the adjusted gross income is $1900 is $225.
c) The independent variable is x and the dependent variable is T.
d) The graph of the linear function over the domain specified in part a is a line that passes through the points (1500, 150) and (52200, 7830).
e) A single filer's adjusted gross income if the tax bill is $3975 is $5200.
b) To find the tax bill for an adjusted gross income of $1900, we just substitute x = 1900 into the function:
T(1900) = 0.15(1900 - 1500) + 150
T(1900) = 0.15(400) + 150
T(1900) = 60 + 150
T(1900) = 210
Therefore, a single filer's tax bill for an adjusted gross income of $1900 is $210.
c) The independent variable is x, which represents the adjusted gross income of a single person. The dependent variable is T, which represents the tax bill.
d) I'm sorry, as a text-based bot, I am unable to provide a visual graph. However, you can plot the graph by taking the domain [1500, 52200] on the x-axis and the calculated tax bill T(x) on the y-axis. Connect the two points (1500, T(1500)) and (52200, T(52200)) with a straight line.
e) To find the adjusted gross income for a tax bill of $3975, we need to solve the equation 3975 = 0.15(x - 1500) + 150 for x. Let's do the calculations:
3975 - 150 = 0.15(x - 1500)
3825 = 0.15(x - 1500)
3825 / 0.15 = x - 1500
25500 = x - 1500
x = 25500 + 1500
x = 27000
Therefore, a single filer's adjusted gross income would be $27,000 if the tax bill is $3975.
b) To find a single filer's tax bill if the adjusted gross income is $1900, we can substitute x = $1900 into the function T(x) and calculate:
T(1900) = 0.15(1900 - 1500) + 150
= 0.15(400) + 150
= 60 + 150
= $210
So, a single filer's tax bill would be $210 if the adjusted gross income is $1900.
c) In this context, the independent variable is x, which represents the adjusted gross income. The dependent variable is T(x), which represents the tax bill. The value of T(x) depends on the value of x.
d) To graph the linear function, we can plot a few points and draw a straight line between them.
Let's choose x = 1500 and x = 52200 as two points within the specified domain:
When x = 1500, T(x) = 0.15(1500 - 1500) + 150 = 150
When x = 52200, T(x) = 0.15(52200 - 1500) + 150 = 7845
Plotting these points on a graph and connecting them with a straight line, we get:
```
|
| *
| *
T | *
|_______|_______________
1500 52200
```
The line represents the linear function T(x) = 0.15(x - 1500) + 150.
e) To find the adjusted gross income if the tax bill is $3975, we can rearrange the function T(x) and solve for x:
0.15(x - 1500) + 150 = 3975
0.15(x - 1500) = 3975 - 150
0.15(x - 1500) = 3825
x - 1500 = 3825 / 0.15
x - 1500 = 25500
x = 25500 + 1500
x = 27000
So, a single filer's adjusted gross income would be $27,000 if the tax bill is $3975.
b) To find a single filer's tax bill if the adjusted gross income is $1900, we can substitute x = 1900 into the function T(x):
T(1900) = 0.15(1900 - 1500) + 150
T(1900) = 0.15(400) + 150
T(1900) = 60 + 150
T(1900) = 210
Therefore, a single filer's tax bill with an adjusted gross income of $1900 would be $210.
c) In this function, the independent variable is x (adjusted gross income) because it is the input into the function. The dependent variable is T(x) (tax bill) because it depends on the value of x.
d) To graph the linear function, we can plot points on a coordinate plane. The domain of the function is [1500, 52200], so we can choose multiple x-values within this range and calculate the corresponding y-values.
For example:
- When x = 1500: T(1500) = 0.15(1500 - 1500) + 150 = 150
- When x = 2500: T(2500) = 0.15(2500 - 1500) + 150 = 300
- When x = 52200: T(52200) = 0.15(52200 - 1500) + 150 = 7905
Plotting these points (1500, 150), (2500, 300), and (52200, 7905) on a graph and drawing a straight line through them will give us the graph of the linear function T(x).
e) To find a single filer's adjusted gross income if the tax bill is $3975, we can set the tax bill T(x) equal to 3975 and solve for x:
3975 = 0.15(x - 1500) + 150
3825 = 0.15(x - 1500)
3825 / 0.15 = x - 1500
25500 = x - 1500
x = 25500 + 1500
x = 27000
Therefore, a single filer's adjusted gross income would be $27,000 if the tax bill is $3975.