Question

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using \(y=f(x)\) to represent a function, an applied problem might use \(C=C(q)\) to represent the cost C of manufacturing qunits of a good. Because of this, the inverse notation \(f^{-1}\) used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as \(C=C(q)\) will be \(q=q(C) .\) So \(C=C(q)\) is a function that represents the cost \(\mathrm{C}\) as a function of the number \(q\) of units manufactured, and \(q=q(C)\) is a function that represents the number \(q\) as a function of the cost \(C\). Illustrate this idea. Income Taxes The function $$ T(g)=4453.05+0.22(g-38,700) $$ represents the 2018 federal income tax \(T\) (in dollars) due for a "single" filer whose modified adjusted gross income is \(g\) dollars, where \(38,700

Short answer

Domain of T(g) = (38,700, 82,500). Range of T(g) = (4453.05, 14089.05). g(T) = \( \frac{T - 4453.05}{0.22} + 38,700 \). Domain of g(T) = (4453.05, 14089.05). Range of g(T) = (38,700, 82,500).

Step-by-step solution

Determine the Domain of T(g)

The domain of a function is the set of all possible input values for which the function is defined. According to the problem statement, the function is defined for a single filer with modified adjusted gross income in the range $$38,700 < g \leq 82,500.$$ Therefore, the domain of the function is $$\text{Domain of } T(g) = (38,700, 82,500).$$

Identify the Range of T(g)

Since the tax due T is an increasing linear function of g, the range will be the set of all values that T(g) takes as g varies over its domain. First, find the lowest and highest values of T(g). For the low end, when $$g = 38,700: \ T(38,700) = 4453.05 + 0.22(38,700 - 38,700) = 4453.05.$$ For the high end, when $$g = 82,500: \ T(82,500) = 4453.05 + 0.22(82,500 - 38,700) = 4453.05 + 9,636 = 14,089.05.$$ Therefore, the range of the function is $$\text{Range of } T(g) = (4453.05, 14089.05).$$

Find g as a Function of T

To find adjusted gross income g as a function of federal income tax T, solve the equation for g in terms of T. Starting from $$T = 4453.05 + 0.22(g - 38,700),$$ isolate g: $$T - 4453.05 = 0.22(g - 38,700) \ \ g - 38,700 = \frac{T - 4453.05}{0.22} \ \ g = \frac{T - 4453.05}{0.22} + 38,700.$$ Thus, the function is $$g = g(T) = \frac{T - 4453.05}{0.22} + 38,700.$$

Identify the Domain and Range of g(T)

The domain of g(T) corresponds to the range of T(g), which we found in Step 2. Therefore, the domain of g(T) is $$\text{Domain of } g(T) = (4453.05, 14,089.05).$$ The range of g(T) corresponds to the domain of T(g), which we found in Step 1. Therefore, the range of g(T) is $$\text{Range of } g(T) = (38,700, 82,500).$$

Key concepts

Independent Variable

Understanding the concept of an independent variable is crucial when studying functions. In simple terms, the independent variable is the variable you are manipulating or choosing values for. In the given exercise, the independent variable is the modified adjusted gross income, denoted by \( g \).
When dealing with functions in application-based problems, you often need to determine this variable before analyzing the relationship it has with the dependent variable. For instance, let's consider the function representing federal income tax, \( T(g) = 4453.05 + 0.22(g - 38,700) \). Here, \( g \) is the independent variable because it represents the adjusted gross income, a value that changes independently.
Identifying the independent variable is the first step in defining the domain of the function, which is the set of all possible values this variable can take. According to the given problem, \( g \) varies between \( 38,700 \) and \( 82,500 \). So, the domain of \( T(g) \) is \( (38,700, 82,500) \).

Dependent Variable

Next up is the dependent variable. As the name suggests, this variable depends on the value chosen for the independent variable. In our function \( T(g) \), the dependent variable is \( T \), which represents the federal income tax due.
This dependency means that the value of the tax is determined by the value of \( g \). When \( g \) changes, \( T \) changes accordingly. For instance, using the formula \( T(g) = 4453.05 + 0.22(g - 38,700) \), if \( g \) is \( 40,000 \), you can calculate the corresponding tax due. This relationship emphasizes the importance of the dependent variable, as it reflects the output of the function based on the input values from the independent variable.
It’s essential to find the range of the dependent variable \( T \), which is the set of all possible values it can take. Calculating the range involves substituting the minimum and maximum values of \( g \) into the function \( T(g) \). With \( g \) between \( 38,700 \) and \( 82,500 \), we find that the range of \( T \) is \( (4453.05, 14,089.05) \).

Domain and Range

The concepts of domain and range are fundamental in understanding how functions operate. The domain is the set of all possible input values for the function, and the range is the set of all possible output values.
In our example, when considering the function \( T(g) = 4453.05 + 0.22(g - 38,700) \), the domain is \( (38,700, 82,500) \) because these are the values \( g \) can take. To find the range, we substitute these domain values into the function to get the output values of \( T \).
To find the inverse function where we express \( g \) in terms of \( T \), we solved for \( g \) to get \( g(T) = \frac{T - 4453.05}{0.22} + 38,700 \). Here, the domain of the inverse function \( g(T) \) is the range of \( T(g) \) which is \( (4453.05, 14,089.05) \). Correspondingly, the range of \( g(T) \) is the domain of \( T(g) \), which is \( (38,700, 82,500) \). Understanding these relationships helps in visualizing how changes in one variable affect the other and in defining the function comprehensively.