​(b)  Given that the income tax due T is an increasing linear function of adjusted gross income​ g, find the range of the function T.

### (b) Range of the Function \( T(g) \):

The function \( T(g) = 4250 + 0.25(g - 30,700) \) is an increasing linear function, meaning as \( g \) increases, \( T(g) \) also increases. To find the range of \( T(g) \), we need to determine the minimum and maximum values of \( T(g) \) over the domain \( g \in [30,700, 74,300] \).

#### 1. Calculate \( T(g) \) at the lower bound \( g = 30,700 \):

\[
T(30,700) = 4250 + 0.25(30,700 - 30,700) = 4250 + 0.25(0) = 4250
\]

#### 2. Calculate \( T(g) \) at the upper bound \( g = 74,300 \):

\[
T(74,300) = 4250 + 0.25(74,300 - 30,700) 
\]

\[
T(74,300) = 4250 + 0.25(43,600) = 4250 + 10,900 = 15,150
\]

### Conclusion:
- **Range:** The range of the function \( T(g) \) is the interval \([4250, 15,150]\).

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

Here are some related questions:

1. How do you interpret the range of \( T(g) \) in terms of income tax?
2. What does the slope of the function \( T(g) \) represent in this context?
3. How would changes in the tax rate affect the range of \( T(g) \)?
4. Can the function \( T(g) \) ever have a negative value? Why or why not?
5. How do you find the inverse of \( T(g) \) and determine its domain and range?

**Tip:** When finding the range of a linear function, evaluating the function at the endpoints of its domain provides the minimum and maximum values.

Explore the range of the function T(g) = 4250 + 0.25(g - 30,700), an increasing linear function representing income tax due based on adjusted gross income. Learn how to calculate the minimum and maximum values of T(g) over the domain g in [30,700, 74,300]. Suitable for advanced high school students and beyond.

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