The inverse notation

fSuperscript negative 1−1

used in a pure mathematics problem is not always used when finding inverses of applied problems. ​ Rather, the inverse of a function such as

Cequals=​C(q)

will be

qequals=​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 comma 70030,700less than or equals≤gless than or equals≤74 comma 30074,300.

​T(g)equals=42504250plus+​0.25(gminus−30 comma 70030,700​)

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Part 1

​(a)  What is the domain of the function​ T?

The problem provides the income tax function \( T(g) = 4250 + 0.25(g - 30,700) \), which is defined for a specific range of adjusted gross income \( g \).

### (a) Domain of the Function \( T(g) \):

The domain of a function refers to all possible input values (in this case, the values of \( g \)) for which the function is defined.

Here, the function \( T(g) \) is defined for:

\[
30,700 \leq g \leq 74,300
\]

### Conclusion:
- **Domain:** The domain of the function \( T(g) \) is the closed interval \([30,700, 74,300]\).

Would you like more details, or do you have any questions?

Here are some related questions:

1. How do you find the range of the function \( T(g) \)?
2. What is the meaning of the slope in the linear function \( T(g) \)?
3. How would you find the inverse function \( g(T) \) for the given problem?
4. How does the function \( T(g) \) change if the tax rate changes?
5. How would the function \( T(g) \) change if the starting amount \( 4250 \) was different?

**Tip:** When interpreting functions, always check the domain and range to ensure the function is correctly applied to the problem context.

Explore the domain of the income tax function T(g) = 4250 + 0.25(g - 30,700). Understand the range of adjusted gross income values for which the function T(g) is defined. Suitable for high school students studying functions and linear equations.

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