Questions: The inverse notation f^-1 used in a pure mathematics problem is not always used when finding inverses of applied problems. Rather, the inverse of a function such as C=C(q) will be q=q(C). The following problem illustrates this idea. In a certain country, the following function represents the income tax T (in dollars) due for a person whose adjusted gross income is g dollars, where 30,650 ≤ g ≤ 74,250. T(g)=4230+0.25(g-30,650) (a) What is the domain of the function T ? The domain of the function T is g ≤ g ≤ .

The inverse notation f^-1 used in a pure mathematics problem is not always used when finding inverses of applied problems. Rather, the inverse of a function such as C=C(q) will be q=q(C). The following problem illustrates this idea.
In a certain country, the following function represents the income tax T (in dollars) due for a person whose adjusted gross income is g dollars, where 30,650 ≤ g ≤ 74,250.
T(g)=4230+0.25(g-30,650)
(a) What is the domain of the function T ?

The domain of the function T is g ≤ g ≤ .

Transcript text: The inverse notation $\mathrm{f}^{-1}$ used in a pure mathematics problem is not always used when finding inverses of applied problems. Rather, the inverse of a function such as $\mathrm{C}=\mathrm{C}(\mathrm{q})$ will be $\mathrm{q}=\mathrm{q}(\mathrm{C})$. The following problem illustrates this idea. In a certain country, the following function represents the income tax T (in dollars) due for a person whose adjusted gross income is g dollars, where $30,650 \leq \mathrm{g} \leq 74,250$. \[ T(g)=4230+0.25(g-30,650) \] (a) What is the domain of the function $T$ ? The domain of the function $T$ is $\{\mathrm{g} \mid$ $\square$ $\leq g \leq$ $\square$ \}.

Solution

Solution Steps

To determine the domain of the function $ T(g) = 4230 + 0.25(g - 30,650) $, we need to identify the range of values that $ g $ can take. The problem states that $ g $ is between 30,650 and 74,250. Therefore, the domain of the function $ T $ is the interval $[30,650, 74,250]$.

Step 1: Identify the Function

The income tax function is given by \[ T(g) = 4230 + 0.25(g - 30650) \] where $ g $ represents the adjusted gross income in dollars.

Step 2: Determine the Domain

According to the problem, the adjusted gross income $ g $ is constrained within the interval \[ 30650 \leq g \leq 74250. \] This means that the function $ T $ is defined for all values of $ g $ within this range.

Step 3: Express the Domain

The domain of the function $ T $ can be expressed in set notation as \[ \{ g \mid 30650 \leq g \leq 74250 \}. \]