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.
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
1 answer