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 (a) e) What is a single filer's adjusted gross income if the tax bill is $3975
a) The domain of this linear function is $1500 \le x \le 52200$.
b) To find the tax bill for an adjusted gross income of $1900, we plug this value into the function:
$T(1900) = 0.15(1900-1500) + 150 = 0.15(400) + 150 = 60 + 150 = 210$.
So, the single filer's tax bill is $210.
c) The independent variable is $x$ (the adjusted gross income), while the dependent variable is $T(x)$ (the tax bill).
d) To graph this linear function over the domain specified in part (a), we first find the coordinates of the endpoints of the domain:
For $x=1500$, we have $T(1500)=0.15(1500-1500)+150=0+150=150$. Thus, the point $(1500,150)$ is on the graph.
For $x=52200$, we have $T(52200)=0.15(52200-1500)+150=0.15(50700)+150=7605+150=7755$. Thus, the point $(52200,7755)$ is on the graph.
We can now plot these two points on a graph and draw a line connecting them:
[asy]
size(250);
import graph;
real xmin=0,xmax=60,ymin=0,ymax=100;
real T(real x) { return 0.15*(x-25) + 25; }
draw(graph(T,25,559),red);
dot((25,25),blue);
dot((559,93.25),blue);
dot((34,28.35),blue);
label("$x$",(xmax,0),S);
label("$y$",(0,ymax),W);
label("$1500$",(25,0),S);
label("$1900$",(34,0),S);
label("$52200$",(559,0),S);
label("$150$",(0,25),W);
label("$210$",(0,28.35),W);
label("$7755$",(0,93.25),W);
xlims(0,600); ylims(0,100);
[/asy]
Note that the graph isn't to scale; however, it does show the relationship between the two variables in the domain specified in part (a).
e) To find the single filer's adjusted gross income when the tax bill is $3975, we set $T(x)=3975$ and solve for $x$:
$3975 = 0.15(x-1500) + 150$
To isolate the term with $x$, we subtract $150$ from both sides:
$3825 = 0.15(x-1500)$
Now, divide both sides by 0.15:
$25500 = x-1500$
Finally, add $1500$ to both sides to solve for $x$:
$x = 25500 + 1500 = 27000$
So, the single filer's adjusted gross income is $27,000.
a) The domain of this linear function is the range of possible values for the adjusted gross income, x. In this case, the domain is from $1500 to $52,200, inclusive.
b) To find a single filer's tax bill if the adjusted gross income is $1900, substitute x = $1900 into the function T(x) and solve for T:
T(x) = 0.15(x - 1500) + 150
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 would be $210 if the adjusted gross income is $1900.
c) In this linear function, the independent variable is the adjusted gross income, x. The dependent variable is the tax bill, T(x). The tax bill depends on the adjusted gross income.
d) To graph the linear function, plot points for different values of x within the specified domain (from $1500 to $52,200). Using the slope-intercept form, the equation of the line is:
T(x) = 0.15(x - 1500) + 150
Plot the points (x, T(x)) and connect them with a straight line:
(x = 1500, T(x) = 150)
(x = 52200, T(x) = 52,200 * 0.15 = 7830)
The graph should show a line starting at (1500, 150) and ending at (52200, 7830). The line represents the tax bill T(x) based on the adjusted gross income x within the specified domain.
e) To find a single filer's adjusted gross income if the tax bill is $3975, we need to solve the equation for x. Rearrange the equation:
0.15(x - 1500) + 150 = 3975
0.15(x - 1500) = 3975 - 150
0.15(x - 1500) = 3825
Divide both sides of the equation by 0.15:
x - 1500 = 3825 / 0.15
x - 1500 = 25500
Add 1500 to both sides of the equation:
x = 25500 + 1500
x = $27,000
Therefore, a single filer's adjusted gross income would be $27,000 if the tax bill is $3975.
a) To find the domain of the function T(x), we need to examine the income range for which the function is defined. In this case, the income is between $1500 and $52,200, inclusive. Therefore, the domain of the function is [1500, 52200].
b) To find the tax bill for an adjusted gross income (x) of $1900, we can substitute x = 1900 into the function T(x) and solve for T(1900):
T(x) = 0.15(x-1500) + 150
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, which represents the adjusted gross income. The dependent variable is T(x), which represents the tax bill. The tax bill depends on the adjusted gross income.
d) To graph the linear function T(x), we plot points on the graph with x-values from the domain [1500, 52200]. Since this is a linear equation, we only need to plot two points and draw a straight line through them.
For example, let's choose two values within the domain, such as x = 1500 and x = 52200:
When x = 1500:
T(x) = 0.15(1500 - 1500) + 150
T(x) = 150
The point (1500, 150) represents the first point on the graph.
When x = 52200:
T(x) = 0.15(52200 - 1500) + 150
T(x) = 7725
The point (52200, 7725) represents the second point on the graph.
Plotting these two points and drawing a straight line through them will give you the graph of the linear function T(x) over the specified domain.
e) To find the adjusted gross income (x) when the tax bill is $3975, we can set T(x) equal to 3975 and solve for x:
3975 = 0.15(x - 1500) + 150
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 $3,975.