(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.