Problem 15 - Algebra for College Students

Question

(a) list the domain and range of the function, (b) form the inverse function \(f^{-1}\), and (c) list the domain and range of \(f^{-1}\). $$ f=\\{(1,5),(2,9),(5,21)\\} $$

Step-by-Step Solution

Answer

Domain of \( f \) is \( \{1, 2, 5\} \), Range is \( \{5, 9, 21\} \). Inverse \( f^{-1} \) is \( \{(5,1), (9,2), (21,5)\} \). Domain of \( f^{-1} \) is \( \{5, 9, 21\} \), Range is \( \{1, 2, 5\} \).

Step 1: Identify the Domain and Range of Function f

The function \( f \) is given as \( \{(1,5),(2,9),(5,21)\} \). The **domain** is the set of all first elements (inputs) of the ordered pairs. Therefore, the domain of \( f \) is \( \{1, 2, 5\} \). The **range** is the set of all second elements (outputs) of the ordered pairs. Therefore, the range of \( f \) is \( \{5, 9, 21\} \).

Step 2: Form the Inverse Function f^{-1}

To find the inverse function \( f^{-1} \), we need to swap the elements of each pair in the set \( f \). The inverse function \( f^{-1} \) is therefore \( \{(5,1), (9,2), (21,5)\} \).

Step 3: Identify the Domain and Range of Inverse Function f^{-1}

For \( f^{-1} \), the **domain** is the set of first elements of the inverse pairs. Thus, the domain of \( f^{-1} \) is \( \{5, 9, 21\} \). The **range** for \( f^{-1} \) is the set of second elements in the inverse pairs, which is \( \{1, 2, 5\} \).

Key Concepts

Inverse Function

An inverse function essentially reverses the roles of the input and output of the original function. For the original function, let's call it "function \( f \)," you're given a set of ordered pairs like \( \{(1,5),(2,9),(5,21)\} \). Each pair displays a number from the domain (input) and its corresponding output (from the range). To find the inverse function, \( f^{-1} \), you simply swap each ordered pair. So, \( (x,y) \) becomes \( (y,x) \).

• Swap ordered pairs: If function \( f \) is \( \{(1,5),(2,9),(5,21)\} \), then \( f^{-1} \) is \( \{(5,1),(9,2),(21,5)\} \).

This means for every output of the original function, \( f \), it becomes the input in \( f^{-1} \). This swapping process is central to understanding what an inverse function is: reversing inputs and outputs.

Ordered Pairs

Ordered pairs are foundational in the study of functions. They are not just numbers thrown together but imply a directed relationship, often represented as \( (x,y) \).
The "first element" of each pair is explained as the domain, or input, and the "second element" as the range, or output.

• The first number is the input (from the domain).
• The second number is the output (from the range).

For example, in the set \( \{(1,5),(2,9),(5,21)\} \), each number on the left—1, 2, and 5—are the inputs being translated into the outputs—5, 9, and 21.When forming an inverse function, these pairs flip, becoming \( \{(5,1),(9,2),(21,5)\} \). Now, the possible "outputs" from the original function become the new "inputs" for the inverse function. Understanding ordered pairs helps grasp the changes that happen in inverse functions.

Mapping

Mapping in functions is about creating a correspondence between the domain and the range. Think of it as a way of showing which inputs get paired with which outputs. This trustful bond is visualized through ordered pairs.In the function \( f = \{(1,5),(2,9),(5,21)\} \):- The number 1 maps to 5,- 2 maps to 9, and- 5 maps to 21.Mapping really matters when you reverse the function to discover \( f^{-1} \). Inverse mapping then shows:- 5 maps back to 1,- 9 maps back to 2, and- 21 maps back to 5.In essence, mapping helps us visualize and understand the relationships that build a function and its inverse. Every mapping explains the function’s behavior and its mirrored operation in the inverse. It's a simple way to see math come to life in functions and inverses.