Problem 16 - 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 \(f^{-1}\). $$ f=\\{(1,1),(4,2),(9,3),(16,4)\\} $$

Step-by-Step Solution

Answer

Domain of \( f = \{1, 4, 9, 16\} \), range of \( f = \{1, 2, 3, 4\} \); \( f^{-1} = \{(1,1), (2,4), (3,9), (4,16)\} \); domain of \( f^{-1} = \{1, 2, 3, 4\} \), range of \( f^{-1} = \{1, 4, 9, 16\}.\)

Step 1: Determine the Domain of the Function

The domain of a function is the set of all possible input values (x-values) of the function. For the given function \( f = \{(1,1),(4,2),(9,3),(16,4)\} \), the domain consists of the first element in each ordered pair: \( \{1, 4, 9, 16\} \).

Step 2: Determine the Range of the Function

The range of a function is the set of all possible output values (y-values) of the function. For the given function \( f = \{(1,1),(4,2),(9,3),(16,4)\} \), the range consists of the second element in each ordered pair: \( \{1, 2, 3, 4\} \).

Step 3: Form the Inverse Function

To find the inverse function \( f^{-1}\), simply swap the x and y values in each pair of the original function. The inverse function \( f^{-1} = \{(1,1), (2,4), (3,9), (4,16)\} \).

Step 4: Determine the Domain of the Inverse Function

The domain of the inverse function \( f^{-1} \) is the set of all x-values of the inverse. These are the y-values of the original function, which are \( \{1, 2, 3, 4\} \).

Step 5: Determine the Range of the Inverse Function

The range of the inverse function \( f^{-1} \) is the set of all y-values of the inverse. These are the x-values of the original function, which are \( \{1, 4, 9, 16\} \).

Key Concepts

Domain of a Function

The domain of a function is crucial for understanding the "input" it allows. Imagine a function like a machine: it requires certain inputs to produce outputs. These inputs are the domain. For instance, consider the function \( f = \{(1, 1), (4, 2), (9, 3), (16, 4)\} \). Here, each ordered pair represents a point where the first number, or "x-value," is part of the domain.

This means the domain of our function \( f \) is \( \{1, 4, 9, 16\} \).

Knowing the domain helps us understand the limits of our function. If you try to plug in a value not within the domain, the function won't be able to give you an answer. That's why it's essential in function analysis.

• Remember: The domain is made up of potential input values.
• Only the first element in each pair contributes to the domain.

Range of a Function

The range of a function is about its "output," or what can be produced from the inputs listed in the domain. After processing each input, the function generates an output. These outputs make up the range.

Using our function \( f = \{(1, 1), (4, 2), (9, 3), (16, 4)\} \), we get the range by listing all the second numbers in each ordered pair: \( \{1, 2, 3, 4\} \).

This range tells us what kind of results to expect from a specific set of inputs.

• Outputs or results are the essence of the range.
• Pay attention to only the second element in each pair for the range.

Understanding the range is key for knowing what outcomes are possible and how they relate to the domain.

Ordered Pairs

Ordered pairs are fundamental to any function. Each pair has a distinct purpose: the first element is the input, or "x-value," and the second is the output, or "y-value." This order is what allows the mapping of specific inputs to specific outputs in a function.

In our function \( f = \{(1, 1), (4, 2), (9, 3), (16, 4)\} \), each ordered pair directly supports identifying the function's domain and range. The inverse function \( f^{-1} = \{(1,1), (2,4), (3,9), (4,16)\} \), simply swaps each pair's elements, creating new input-output relationships.

Understanding ordered pairs helps in visualizing and predicting function behaviors, as well as forming inverse functions.

• Ordered pairs display the connection between inputs and outputs.
• The order (input, output) is crucial to maintaining function structure.

Mastering ordered pairs is essential for grasping how functions, and their inverses, interact and communicate numbers.