Income Taxes The function T(g)=5081.25+0.25(g-36,900)...

Income Taxes The function $$ T(g)=5081.25+0.25(g-36,900) $$ represents the 2014 federal income tax $T$ (in dollars) due for a "single" filer whose modified adjusted gross income is g dollars, where $36,900 \leq g \leq 89,350$. (a) What is the domain of the function $T ?$ (b) Given that the tax due $T$ is an increasing linear function of modified adjusted gross income $g,$ find the range of the function $T$. (c) Find adjusted gross income $g$ as a function of federal income tax $T .$ What are the domain and the range of this function?

Income Taxes The function
$$
T(g)=5081.25+0.25(g-36,900)
$$
represents the 2014 federal income tax $T$ (in dollars) due for a "single" filer whose modified adjusted gross income is g dollars, where $36,900 \leq g \leq 89,350$.
(a) What is the domain of the function $T ?$
(b) Given that the tax due $T$ is an increasing linear function of modified adjusted gross income $g,$ find the range of the function $T$.
(c) Find adjusted gross income $g$ as a function of federal income tax $T .$ What are the domain and the range of this function?

Key Concepts

Inverse Function

An inverse function reverses the roles of inputs and outputs. If you have a function f such that y = f(x), the inverse function f?¹ is defined by x = f?¹(y). Determining an inverse function involves solving for the original input variable in terms of the output variable. It is important to consider the domain and range of both the original function and its inverse, ensuring that operations are valid and meaningful within the context of the problem.

Domain

The domain of a function includes all the possible input values for which the function is defined. In the context of functions representing real-world scenarios, such as tax calculations, the domain sets the boundaries within which the function is applicable. This is important because it ensures that the function is used only for meaningful and valid inputs, reflecting the constraints of the problem or situation at hand.

Range

The range of a function is the set of all possible output values obtained from the function when its domain is considered. For increasing linear functions, once the inputs are restricted by the domain, the outputs naturally correspond to an interval that is determined by the extreme values produced at the boundaries of the domain. Understanding the range is crucial for interpreting the results and predicting the behavior of the function across its allowable inputs.

Linear Function

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. Its graph is a straight line, and its rate of change (slope) is constant. This structure provides straightforward methods for identifying the domain, range, and behavior of the function, and plays a significant role in applications where relationships are proportional or exhibit constant rates of change.

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