Problem 88 - 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 \(\boldsymbol{f}^{-1}\) $$ \begin{array}{cccc} x & 0 & 1 & 2 \\ f(x) & 1 & 2 & 4 \end{array} $$

Step-by-Step Solution

Answer

The table for \( f^{-1}(x) \) is: \begin{array}{cccc} x & 1 & 2 & 4 \\ f^{-1}(x) & 0 & 1 & 2 \\ \end{array}. Domains: \( f \) is \{0, 1, 2\}, \( f^{-1} \) is \{1, 2, 4\}; Ranges: \( f \) is \{1, 2, 4\}, \( f^{-1} \) is \{0, 1, 2\}.

Step 1: Understanding the Table for f(x)

The table provided shows values for the function \( f(x) \). Specifically, we have the pairs \( (0, 1), (1, 2), (2, 4) \). This means that \( f(0) = 1 \), \( f(1) = 2 \), and \( f(2) = 4 \).

Step 2: Setting Up the Inverse Table

The inverse function \( f^{-1}(x) \) essentially reverses the roles of the inputs and outputs. This means that each \( f(x) \) value from our original table becomes an input for the inverse function, and each original \( x \) value becomes the output. We start building the table for \( f^{-1}(x) \) by reversing these pairs: \( (1, 0), (2, 1), (4, 2) \).

Step 3: Completing the Table for f^{-1}(x)

Using the reversed pairs from the previous step, we populate the table for \( f^{-1}(x) \):\[ \begin{array}{cccc}x & 1 & 2 & 4 \ f^{-1}(x) & 0 & 1 & 2 \\end{array} \]

Step 4: Identifying the Domain and Range of f(x)

The domain of \( f(x) \) is the set of original \( x \) values: \{0, 1, 2\}. The range of \( f(x) \) is the set of corresponding \( f(x) \) values: \{1, 2, 4\}.

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

For \( f^{-1}(x) \), the domain is the set of \( x \) values it can take, which are the \( f(x) \) values from the original function: \{1, 2, 4\}. The range is the set of \( f^{-1}(x) \) outputs, the original \( x \) values: \{0, 1, 2\}.

Key Concepts

Domain and Range

Determining the domain and range of a function and its inverse is essential for understanding what inputs a function can accept, and what outputs it will produce. In the given problem, determine both these aspects for the original function \( f(x) \) and its inverse \( f^{-1}(x) \).

**For the function \( f(x) \):**
- **Domain:** This represents all possible \( x \) values that \( f(x) \) can accept. For our function \( f(x) = \{0, 1, 2\} \).
- **Range:** These are all the possible outputs, the \( y \) values that \( f(x) \) can produce. In this example, the range is \( \{1, 2, 4\} \).

**For the inverse function \( f^{-1}(x) \):**
- **Domain:** This is the set of values that \( f^{-1}(x) \) can take as input. It coincides with the range of \( f(x) \), giving us \( \{1, 2, 4\} \).
- **Range:** Outputs of the inverse function correspond to the domain of \( f(x) \), therefore, \( \{0, 1, 2\} \).
Understanding domains and ranges is pivotal as it helps identify the boundaries of functions and ensures correctness in solving mathematical problems.

Function Tables

Function tables are a convenient way to visualize how a function maps inputs to outputs. Working with these tables helps in understanding the behavior and relationship between the values of the function and its inverse. Consider the function table provided:

\[ \begin{array}{ccc} x & 0 & 1 & 2 \ f(x) & 1 & 2 & 4 \end{array} \]

In this table:
- The first row displays the input values \( x \), and these define the domain of \( f(x) \).
- The second row shows the corresponding outputs \( f(x) \), presenting the range of \( f(x) \).

The role of the function table becomes even more apparent when reversing the input-output roles to form the inverse. By switching positions of \( x \) and \( f(x) \), you create \( f^{-1}(x) \):

\[ \begin{array}{ccc} x & 1 & 2 & 4 \ f^{-1}(x) & 0 & 1 & 2 \end{array} \]

This reversal shows how each output of the original function becomes an input in the inverse, and vice versa. Function tables make it straightforward to visualize these pivotal relationships.

Step-by-Step Solutions

Step-by-step solutions are invaluable tools in learning math concepts like inverse functions. They not only guide you through the process but also demonstrate the logical sequence necessary to solve complex problems. For this problem, understanding and creating an inverse table involves several clear steps.

First, interpret the given table for \( f(x) \) with assigned input-output pairs: \( (0, 1), (1, 2), (2, 4) \). This establishes initial conditions and helps define the function \( f(x) \).

Next, transition to the inverse function \( f^{-1}(x) \) by flipping these pairs. Here, each \( f(x) \) value becomes the new input, while the old \( x \) values serve as outputs to form \( (1, 0), (2, 1), (4, 2) \).

Complete the table for \( f^{-1}(x) \) neatly with these entries, verifying that all values correctly match their reversed relationships. Finally, identify the domains and ranges for both functions to solidify understanding. This step-by-step approach ensures clarity and precision, allowing complex mathematical concepts to become more accessible and less intimidating.