Problem 85 - College Algebra with Modeling and Visualization

Question

Use the table for \(f(x)\) to find a table for \(\boldsymbol{f}^{-1}(\boldsymbol{x})\). Identify the domains and ranges of \(\boldsymbol{f}\) and \(\boldsymbol{f}^{-1}\) $$ \begin{array}{rrrr} x & 1 & 2 & 3 \\ f(x) & 5 & 7 & 9 \end{array} $$

Step-by-Step Solution

Answer

The table for \(f^{-1}(x)\) is \(f^{-1}(x)=\{(5,1),(7,2),(9,3)\}\). The domain of \(f\) is \(\{1,2,3\}\), and the range is \(\{5,7,9\}\). The domain of \(f^{-1}\) is \(\{5,7,9\}\), and the range is \(\{1,2,3\}\).

Step 1: Understand the Given Table

The table shows values of the function \(f(x)\). For each \(x\), \(f(x)\) is given. Specifically, \(f(1) = 5\), \(f(2) = 7\), and \(f(3) = 9\).

Step 2: Define the Inverse Function

To find \(f^{-1}(x)\), we switch the \(x\) values with the \(f(x)\) values. This is because for the inverse function, we want \(f^{-1}(f(x)) = x\). Thus, we are looking for what \(x\) value corresponds to a given \(f(x)\).

Step 3: Construct the Inverse Table

Create a table for \(f^{-1}(x)\) by swapping the columns from the original table:- When \(f(x) = 5\), \(x = 1\), so \(f^{-1}(5) = 1\).- When \(f(x) = 7\), \(x = 2\), so \(f^{-1}(7) = 2\).- When \(f(x) = 9\), \(x = 3\), so \(f^{-1}(9) = 3\).The inverse table is:\[\begin{array}{ccc} f^{-1}(x) & 5 & 7 & 9 \ x & 1 & 2 & 3 \\end{array}\]

Step 4: Identify the Domain and Range of \(f\)

For the function \(f(x)\), the domain is all \(x\) values from the original table: \(\{1, 2, 3\}\). The range is all \(f(x)\) values: \(\{5, 7, 9\}\).

Step 5: Identify the Domain and Range of \(f^{-1}\)

For the inverse function \(f^{-1}(x)\), the domain is the range of the original function \(f(x)\): \(\{5, 7, 9\}\). The range of \(f^{-1}(x)\) is the domain of the original function \(f(x)\): \(\{1, 2, 3\}\).

Key Concepts

Domain and Range

Understanding the concept of domain and range is crucial for working with functions and their inverses. The **domain** of a function is the set of all possible input values (often represented as 'x' values) that the function can accept. For example, in the original function \(f(x)\) with the table provided, the domain is \(\{1, 2, 3\}\). These are the values 'x' takes in the given table.
The **range**, on the other hand, refers to the set of all possible output values (often represented as 'f(x)' or 'y' values). In our original function \(f(x)\), the range is \(\{5, 7, 9\}\), which are the values that 'f(x)' can take according to the table.
For the inverse function \(f^{-1}(x)\), the domain and range swap roles. Thus, the domain of \(f^{-1}(x)\) is \(\{5, 7, 9\}\) and the range is \(\{1, 2, 3\}\). This switching highlights an important relationship between functions and their inverses.

Function Tables

Function tables are an excellent tool for visualizing how functions work, particularly when dealing with inverse functions. A typical function table lists input values in one column and their corresponding output values in another.
In the provided exercise, the function table for \(f(x)\) is:

• x: 1, 2, 3
• f(x): 5, 7, 9

Each row specifies one pair of 'x' and 'f(x)' values. By swapping these input-output pairs, you create a table for \(f^{-1}(x)\).
This reverse process involves:

• Taking each output value \(f(x)\) (i.e., 5, 7, 9) of the original function to act as inputs \(f^{-1}(x)\) in the inverse table.
• Identifying corresponding input \(x\) values (i.e., 1, 2, 3) as the outputs.

This allows us to clearly see the operation of the inverse function.

Switching x and y values

The process of finding an inverse function essentially involves switching 'x' and 'y' values. In the context of a table, this means that 'x' values of the original function become the 'y' values of the inverse function, and vice versa.
For the given function \(f(x)\) with values \(\{(1, 5), (2, 7), (3, 9)\}\), the inverse function \(f^{-1}(x)\) is obtained by swapping each pair:

• The '5' from \(f(x)\) mapping back to '1' in \(f^{-1}(x)\)
• The '7' mapping to '2'
• The '9' mapping to '3'

This swap is foundational in understanding inverse functions. The relationship between the original function and its inverse can be seen as mirroring across the line \(y = x\), reinforcing the idea of swapping 'x' and 'y' roles. Understanding this relationship helps in visualizing and interpreting inverse functions effectively.