The domain of a function T , where T ( g ) = 4453.05 + 0.22 ( g − 38 , 700 ) represents the 2018 federal income tax T (in dollars) due for a “single” filer, whose modified adjusted gross income is g dollars, where 38 , 700 ≤ g ≤ 82 , 500 .
The domain of a function T , where T ( g ) = 4453.05 + 0.22 ( g − 38 , 700 ) represents the 2018 federal income tax T (in dollars) due for a “single” filer, whose modified adjusted gross income is g dollars, where 38 , 700 ≤ g ≤ 82 , 500 .
The domain of a function T, where T(g)=4453.05+0.22(g−38,700) represents the 2018 federal income tax T(in dollars) due for a “single” filer, whose modified adjusted gross income is g dollars, where 38,700≤g≤82,500.
(a)
Expert Solution
Answer to Problem 93AYU
Solution:
The domain of function T is [38700,82500].
Explanation of Solution
Given information:
T(g)=4453.05+0.22(g−38,700) represents the 2018 federal income tax T(in dollars) due for a “single” filer, whose modified adjusted gross income is g dollars, where 38,700≤g≤82,500.
Explanation:
The values of g works as an input to the function T
Therefore, the domain of T is [38700,82500]
(b)
To determine
The range of the function T, where T is the tax due is an increasing linear function of modified adjusted gross income g.
(b)
Expert Solution
Answer to Problem 93AYU
Solution:
The range of the function T is [4453.05,14089.05]
Explanation of Solution
Given information:
T(g)=4453.05+0.22(g−38,700) represents the 2018 federal income tax T(in dollars) due for a “single” filer, whose modified adjusted gross income is g dollars, where 38,700≤g≤82,500.
Explanation:
The tax due T is an increasing linear function of modified adjusted gross income g.
Therefore, the range of T(g) will have the lowest value at g=38,700.
Thus, substitute g=38,700 in equation of T(g)
⇒T(38,700)=4453.05+0.22(38,700−38,700)
⇒T(38,700)=4453.05+0.22(0)
⇒T(38,700)=4453.05+0
⇒T(38,700)=4453.05
The maximum value of T(g) occur at g=82,500
Thus, substitute g=82,500 in equation of T(g)
⇒T(82,500)=4453.05+0.22(82,500−38,700)
⇒T(82,500)=4453.05+0.22(43800)
⇒T(82,500)=4453.05+9636
⇒T(82,500)=14089.05
Thus, the range of T(g) is [4453.05,14089.05]
(c)
To determine
The adjusted gross income g as a function of federal income tax T. Determine the domain and the range of this function.
(c)
Expert Solution
Answer to Problem 93AYU
Solution:
1) The adjusted gross income function is g(T)=T+4060.950.22.
2) The domain of T(g) is {T(g)|4453.05≤T(g)≤14089.05} and range is {g|38,700≤g≤82,500}.
Explanation of Solution
Given information:
T(g)=4453.05+0.22(g−38,700) represents the 2018 federal income tax T(in dollars) due for a “single” filer, whose modified adjusted gross income is g dollars, where 38,700≤g≤82,500.
Explanation:
Let T(g)=4453.05+0.22(g−38,700)
To find the gross income function g, solve the equation T(g)=4453.05+0.22(g−38,700) for variable g.
⇒T=4453.05+0.22g−0.22(38,700)
⇒T=4453.05+0.22g−8514
⇒T=0.22g−4060.95
⇒0.22g=T+4060.95
⇒g(T)=T+4060.950.22
Functions T(g) and g(T) are inverses of each other.
From part (a) and (b), the domain of T(g) is [38700,82500] and the range of T(g) is [4453.05,14089.05]
Thus, domain of g(T) = range of T(g)=[4453.05,14089.05] and range of g(T) = domain of T(g)= [38700,82500]
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