Problem 17 - 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 \(f^{-1}\). $$ f=\\{(0,0),(2,8),(-1,-1),(-2,-8)\\} $$

Step-by-Step Solution

Answer

Domain and range of \(f\): \(\{0, 2, -1, -2\}\) and \(\{0, 8, -1, -8\}\). Inverse: \{(0,0),(8,2),(-1,-1),(-8,-2)\}. Domain and range of \(f^{-1}\): \(\{0, 8, -1, -8\}\) and \(\{0, 2, -1, -2\}\).

Step 1: Identify the domain and range of the function

To identify the domain of the function, we list all the x-values from each of the ordered pairs. Thus, the domain of the function \(f\) is \(\{0, 2, -1, -2\}\). The range of the function is derived from listing all the y-values corresponding to each x-value, giving us \(\{0, 8, -1, -8\}\).

Step 2: Form the inverse function \(f^{-1}\)

To find the inverse function \(f^{-1}\), we swap the x and y values in each of the ordered pairs of the original function. Thus, the inverse function \(f^{-1}\) becomes \{(0,0),(8,2),(-1,-1),(-8,-2)\}.

Step 3: List the domain and range of the inverse function

For \(f^{-1}\), the domain is derived from the range of the original function, resulting in \(\{0, 8, -1, -8\}\). Similarly, the range of \(f^{-1}\) is derived from the domain of the original function, thus \(\{0, 2, -1, -2\}\).

Key Concepts

Domain and Range

The domain and range are fundamental aspects of understanding a function. They help us know which inputs give valid outputs and provide insight into the overall behavior of the function.
The **domain** of a function is the set of all possible input values (often represented by x-values) that the function can accept. In our function \( f = \{(0,0), (2,8), (-1,-1), (-2,-8)\} \), the domain is determined by extracting the first element of each pair. These are the x-values: \ \[ \{0, 2, -1, -2\}\]
The **range** is the set of all output values (y-values) that the function produces. By examining the second element in each of the given ordered pairs, we find the range \ \[\{0, 8, -1, -8\}\]. It's crucial to grasp both domain and range as they tell us about the function's behavior over all considered values.

Ordered Pairs

Ordered pairs are used to define relations and functions within mathematics. Each pair consists of two elements: an x-value and a y-value. These pairs can help us visually map a function and show the relationship between inputs and outputs.
In the function \( f = \{(0,0), (2,8), (-1,-1), (-2,-8)\} \), each ordered pair is an explicit assignment where the first coordinate is from the domain, and the second from the range.
The unique property of an ordered pair is that the sequence matters:

• (0,0) means the output for input 0 is 0.
• (2,8) means the output for input 2 is 8.
• The reverse, (8,2), will mean that the output for 8 is 2 if we swap for the inverse function.

Understanding ordered pairs is key to swapping x and y values to find an inverse of a function.

Function Properties

Functions have intrinsic properties that make them unique. Grasping these properties allow us to understand the workings of functions better and to manipulate them effectively.
A key property that is vital in building an inverse function is that a function must map each input to exactly one output. This ensures that every element of the domain corresponds to only one element in the range.
To find an **inverse function**, denoted by \( f^{-1} \), you swap each ordered pair in the function
\( f = \{(0,0), (2,8), (-1,-1), (-2,-8)\} \) to get \ \[ f^{-1} = \{(0,0), (8,2), (-1,-1), (-8,-2)\} .\]
Now, the domain of \( f^{-1} \) becomes the range of \( f \), which is \\{0, 8, -1, -8\}, and the range of \( f^{-1} \) becomes the domain of \( f \), which is \ \{0, 2, -1, -2\}.
This swapping showcases another function property: called the symmetry between a function and its inverse. It helps in visualizing the input/output relationship as a mirrored inversion.