Problem 19 - Intermediate Algebra

Question

(a) list the domain and range of the given function, (b) form the inverse function, and (c) list the domain and range of the inverse function. $$f=\\{(-2,-1),(-1,1),(0,5),(5,10)\\}$$

Step-by-Step Solution

Answer

Domain of \(f\): \(-2, -1, 0, 5\); Range of \(f\): \(-1, 1, 5, 10\). Inverse: \{(-1, -2), (1, -1), (5, 0), (10, 5)\}. Domain of \(f^{-1}\): \(-1, 1, 5, 10\); Range of \(f^{-1}\): \(-2, -1, 0, 5\).

Step 1: Determine Domain and Range of the Original Function

The domain of a function is the set of all possible input values (x-values). List all the first elements in each ordered pair: \(-2, -1, 0, 5\). The range is the set of all possible output values (y-values). List all the second elements in each ordered pair: \(-1, 1, 5, 10\). Thus, the domain of \(f\) is \(-2, -1, 0, 5\) and the range is \(-1, 1, 5, 10\).

Step 2: Form the Inverse Function

To form the inverse function, swap each ordered pair so that the first element becomes the second and the second element becomes the first. Change each pair: - From \((-2, -1)\) to \((-1, -2)\)- From \((-1, 1)\) to \((1, -1)\)- From \((0, 5)\) to \((5, 0)\)- From \((5, 10)\) to \((10, 5)\)The inverse function \(f^{-1}\) is \{(-1, -2), (1, -1), (5, 0), (10, 5)\}.

Step 3: Determine Domain and Range of the Inverse Function

The domain of the inverse function is the set of all possible input values for \(f^{-1}\). These are the same as the outputs (range) of the original function: \(-1, 1, 5, 10\).The range of the inverse function is the set of all possible output values for \(f^{-1}\). These are the same as the inputs (domain) of the original function: \(-2, -1, 0, 5\).Thus, the domain of \(f^{-1}\) is \(-1, 1, 5, 10\) and the range is \(-2, -1, 0, 5\).

Key Concepts

domain of a function

Understanding the domain of a function is a fundamental concept in mathematics, particularly in the study of functions. The domain is essentially the set of all possible input values (commonly x-values) that a function can accept.
If you imagine a function as a machine, the domain represents all the different parts you are allowed to feed into it. In our given function \( f = \{(-2,-1),(-1,1),(0,5),(5,10)\} \), the domain is determined by examining the first value of each ordered pair.
Therefore, for this function, the domain is \(-2, -1, 0, 5\).

• This means any of these x-values can be plugged into the function to produce an output.
• It's important to identify the domain when working with functions, as it tells us which values are valid inputs and ensures the function operates within its intended scope.

By understanding the domain, we avoid errors like trying to calculate square roots of negative numbers, or divide by zero, which are not possible within real numbers.

range of a function

The range of a function is akin to its output possibilities – these are the values you might get out of the function after providing inputs within its domain. If the domain is the left side of a function's equation, the range is the right side.
For the function \( f = \{(-2,-1),(-1,1),(0,5),(5,10)\} \), the range consists of all the second values in the ordered pairs. So here, the range is \(-1, 1, 5, 10\).

• These values are the possible outputs the function can yield.
• Evaluating the range helps us understand the function's behavior, especially when translated into real-world scenarios, such as computing temperatures or speeds.

Recognizing the range is critical as it gives clarity on the extent of possible results the function can provide, thus aiding in effectively solving mathematical and applied problems.

ordered pairs

Ordered pairs are integral to understanding the relationship between different sets of data in mathematics. An ordered pair is a set of two elements where the order of those elements is significant.
Typically seen as \((x, y)\) in the context of functions, the first element (x) represents the input from the domain, and the second element (y) represents the output that falls within the range. In our exercise with the function \( f = \{(-2,-1),(-1,1),(0,5),(5,10)\} \), each pair is ordered by its x-value followed by its y-value.

• This organization is crucial because swapping the order changes the meaning and application of the pair, as illustrated by the formation of the inverse function where each pair flips to become \((y, x)\).
• Ordered pairs are foundational in mapping functions and understanding how inputs are directly tied to outputs, which is essential in graphing and analyzing data relationships.

Therefore, understanding ordered pairs ensures proper function definition, accuracy in data interpretation, and can aid in efficiently resolving various algebraic problems.