Problem 43 - Calculus

Question

Write a table of values for \(f^{-1},\) where \(f\) is as given below. The domain of \(f\) is the integers from 1 to \(7 .\) State the domain of \(f^{-1}\). $$\begin{array}{c|c|c|c|c|c|c|c}\hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\hline f(x) & 3 & -7 & 19 & 4 & 178 & 2 & 1 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

\( f^{-1}(y) \) maps \( 3 \to 1, -7 \to 2, 19 \to 3, 4 \to 4, 178 \to 5, 2 \to 6, 1 \to 7 \). Domain of \( f^{-1} \) is \( \{3, -7, 19, 4, 178, 2, 1\} \).

Step 1: Understand the Function

The given function is defined as a table where for each input \( x \) (ranging from 1 to 7), there is an output \( f(x) \). Here, you interpret each value and notice that we have a mapping from the integers 1 to 7 to the outputs \( 3, -7, 19, 4, 178, 2, \) and \( 1 \).

Step 2: Identify the Inverse

The inverse function \( f^{-1} \) reverses the role of the input and output. Each output value of \( f \) becomes the input for \( f^{-1} \) and vice versa. For example, since \( f(1) = 3 \), in the inverse \( f^{-1}(3) = 1 \). Apply this logic to all values.

Step 3: Write the Table for \( f^{-1} \)

Create a new table where the first row contains the output values from the original function \( f \) (which now become inputs for \( f^{-1} \)), and the second row contains the corresponding original inputs from \( f \). The table will look like this: \[\begin{array}{c|c|c|c|c|c|c|c}\hline y & 3 & -7 & 19 & 4 & 178 & 2 & 1 \ \hline f^{-1}(y) & 1 & 2 & 3 & 4 & 5 & 6 & 7 \ \hline\end{array}\]

Step 4: Determine the Domain of \( f^{-1} \)

The domain of \( f^{-1} \) is the set of all unique values from the original \( f(x) \) outputs. From the table, \( f \) outputs the values \( 3, -7, 19, 4, 178, 2, \) and \( 1 \). Therefore, the domain of \( f^{-1} \) is \( \{3, -7, 19, 4, 178, 2, 1\} \).

Key Concepts

Function Table

A function table is an organized way of displaying the input-output pairs of a function. It allows us to visually grasp the relationship between each input, often represented as \( x \), and its corresponding output, represented as \( f(x) \). Function tables can simplify complex ideas by showing how particular inputs connect to their outputs in a clear and concise manner.

To illustrate, consider the example from the exercise where the function \( f \) is defined for integers from 1 to 7. The function table looks like this:

• \( x = 1 \) maps to \( f(x) = 3 \)
• \( x = 2 \) maps to \( f(x) = -7 \)
• \( x = 3 \) maps to \( f(x) = 19 \)
• \( x = 4 \) maps to \( f(x) = 4 \)
• \( x = 5 \) maps to \( f(x) = 178 \)
• \( x = 6 \) maps to \( f(x) = 2 \)
• \( x = 7 \) maps to \( f(x) = 1 \)

Through this table, we clearly see how each input is paired with its respective output, which is crucial when determining inverse functions.

Domain of a Function

The domain of a function refers to the complete set of possible input values, often represented by \( x \), that can be used in a function. In other words, it delineates what can go into the function to produce a legitimate result. When dealing with inverse functions, understanding the domain becomes even more significant.

For instance, in our exercise, the domain of the original function \( f \) is given as the integers from 1 to 7. This implies that the function could accept any integer within this range as an input and give a valid output. For the inverse function \( f^{-1} \), the concept flips. The domain of the inverse becomes the set of all possible outputs from the original function \( f \). Therefore, for \( f^{-1} \), the domain is \( \{3, -7, 19, 4, 178, 2, 1\} \).

When working with inverse functions, always reciprocate roles between the inputs and outputs. This ensures you correctly determine domains and maintain function validity.

Integer Mappings

Integer mappings involve the pairing of integer inputs with their corresponding outputs within a function. A mapping shows how each input integer is uniquely linked to an output, providing a complete picture of the function's operation on integers.

In the context of the exercise, consider the mapping provided:

• 1 maps to 3
• 2 maps to -7
• 3 maps to 19
• 4 maps to 4
• 5 maps to 178
• 6 maps to 2
• 7 maps to 1

Through these mappings, we can easily visualize the directional flow of the function from each integer input to its corresponding output. It's this mapping that the inverse function \( f^{-1} \) reverses, effectively swapping inputs and outputs, thus demonstrating the fundamental principle of inverse functions.