Problem 87 - 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 \(\boldsymbol{f}^{-1}\) $$ \begin{array}{cccc} x & 0 & 2 & 4 \\ f(x) & 0 & 4 & 16 \end{array} $$

Step-by-Step Solution

Answer

Inverse table: \(\begin{array}{cccc} f^{-1}(x) & 0 & 4 & 16 \\ x & 0 & 2 & 4 \end{array}\). Domains: \(f(x)\) - \{0, 2, 4\}, \(f^{-1}(x)\) - \{0, 4, 16\}. Ranges: \(f(x)\) - \{0, 4, 16\}, \(f^{-1}(x)\) - \{0, 2, 4\}.

Step 1: Understanding the Problem

We are given a table that shows values of a function \( f(x) \) and need to construct a table for its inverse \( f^{-1}(x) \). An inverse function, by definition, swaps the roles of inputs and outputs. Hence, wherever \( f(x) = y \), we should have \( f^{-1}(y) = x \).

Step 2: Creating the Table for \( f^{-1}(x) \)

To create the table for \( f^{-1}(x) \), simply switch the columns of \( x \) and \( f(x) \) from the original table. Thus, the inverse table is:\[\begin{array}{cccc} f^{-1}(x) & 0 & 4 & 16 \ x & 0 & 2 & 4\end{array}\]

Step 3: Identifying the Domain and Range of \( f(x) \)

The domain of \( f(x) \) consists of its input values, which are \(0, 2,\) and \(4\). The range of \( f(x) \) consists of its output values, which are \(0, 4,\) and \(16\).

Step 4: Identifying the Domain and Range of \( f^{-1}(x) \)

The domain of \( f^{-1}(x) \) is the same as the range of \( f(x) \), which is \(0, 4,\) and \(16\). The range of \( f^{-1}(x) \) is the same as the domain of \( f(x) \), which is \(0, 2,\) and \(4\).

Key Concepts

Domain and Range

When studying functions, the concepts of domain and range help us understand where a function is defined and what possible output values it can produce. The domain of a function is essentially the set of all possible input values, often represented by the variable \( x \). For example, in the function table above, the domain of \( f(x) \) is \( \{0, 2, 4\} \). These are all the \( x \)-values you can put into the function to get an output.

On the other hand, the range of a function includes all possible output values, which are represented by \( f(x) \) in the given table. For the function \( f(x) \), the range is \( \{0, 4, 16\} \). This range signifies all the values \( f(x) \) can take as outputs when applied to each value in the domain.

For inverse functions like \( f^{-1}(x) \), these roles switch: the domain of the inverse is the range of the original function and vice versa. So, for \( f^{-1}(x) \), the domain becomes \( \{0, 4, 16\} \), and the range switches to \( \{0, 2, 4\} \). Understanding this swapping is crucial to mastering inverse functions.

Function Table

A function table is a simple but powerful tool to understand the relationship between inputs and outputs in a function. In this table setup, we list possible values of \( x \) and the corresponding values of \( f(x) \). It's this relationship that defines the function's behavior. For instance, consider the table for \( f(x) \) given in the exercise:

• The input \( x = 0 \) gives output \( f(x) = 0 \)
• The input \( x = 2 \) gives output \( f(x) = 4 \)
• The input \( x = 4 \) gives output \( f(x) = 16 \)

When constructing the table for the inverse function \( f^{-1}(x) \), you simply switch each ordered pair, reflecting their inverse relationship. This means that each \( x \) becomes an output, and each \( f(x) \) turns into the corresponding input, like so:

• The input \( f^{-1}(x) = 0 \) gives output \( x = 0 \)
• The input \( f^{-1}(x) = 4 \) gives output \( x = 2 \)
• The input \( f^{-1}(x) = 16 \) gives output \( x = 4 \)

This switch highlights the inverse nature of these functions, allowing us to solve them easily.

Function Notation

Understanding function notation is crucial for simplifying how we describe and execute the operations of a function. In mathematics, function notation like \( f(x) \) is used to denote a function named \( f \) with \( x \) being the variable input. This notation helps us easily handle and communicate complex relationships without cumbersome wording.

• \( f(x) = 4 \) tells us that input \( x \) results in an output of 4 through function \( f \).
• \( f(y) \) would mean we're working with some input \( y \) with the function \( f \), emphasizing that the input changes.

Inverse function notation, like \( f^{-1}(x) \), indicates the inverse process of the function \( f \). This notation doesn't mean "one over \( f \)," instead it refers to the function that reverses the effect of \( f \). If \( f(x) = y \), then \( f^{-1}(y) = x \). For instance:

• \( f^{-1}(4) = 2 \) implies an original function with input \( x = 2 \), resulting in output \( y = 4 \).

Using these notations efficiently unlocks a deeper understanding of both functions and inverse functions, as they help succinctly represent mathematical operations.