Problem 17 - 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. $$f=\\{(1,3),(2,6),(3,11),(4,18)\\}$$

Step-by-Step Solution

Answer

Domain of f: {1, 2, 3, 4}; Range of f: {3, 6, 11, 18}; f^{-1}: {(3,1),(6,2),(11,3),(18,4)}; Domain of f^{-1}: {3, 6, 11, 18}; Range of f^{-1}: {1, 2, 3, 4}.

Step 1: Identify the Domain of f

The domain of a function is the set of all possible input values. For the function \( f = \{(1,3),(2,6),(3,11),(4,18)\} \), the domain consists of the first elements from each pair. Thus, the domain of \( f \) is \( \{1, 2, 3, 4\} \).

Step 2: Identify the Range of f

The range of a function is the set of all possible output values. For the function \( f = \{(1,3),(2,6),(3,11),(4,18)\} \), the range consists of the second elements from each pair. Thus, the range of \( f \) is \( \{3, 6, 11, 18\} \).

Step 3: Construct the Inverse Function

To find the inverse of a function, swap each pair’s elements. For \( f = \{(1,3),(2,6),(3,11),(4,18)\} \), the inverse will be \( f^{-1} = \{(3,1),(6,2),(11,3),(18,4)\} \).

Step 4: Identify the Domain of f^{-1}

The domain of the inverse function \( f^{-1} \) is the range of the original function \( f \). Thus, the domain of \( f^{-1} \) is \( \{3, 6, 11, 18\} \).

Step 5: Identify the Range of f^{-1}

The range of the inverse function \( f^{-1} \) is the domain of the original function \( f \). Thus, the range of \( f^{-1} \) is \( \{1, 2, 3, 4\} \).

Key Concepts

Domain of a Function

In algebra, the domain of a function is crucial to understand. It represents all the possible input values that the function can accept without causing any errors or undefined behavior. For instance, if we have the function \( f = \{(1,3),(2,6),(3,11),(4,18)\} \), determining the domain means identifying all the "x" values or the first elements of each ordered pair. In this case, the domain is \( \{1, 2, 3, 4\} \).

The domain is foundational because it sets the limits on what can be plugged into our function. Consider it as the starting point of a journey that will reach specific destinations defined by the range. When dealing with functions, especially those more complex than simple sets of ordered pairs, it's also essential to check for values that could make the function undefined, like dividing by zero or taking square roots of negative numbers. However, in this simple example, the only restriction is the first elements listed in our set.

Range of a Function

The range of a function is equally important. It outlines all possible outputs or results that come from substituting each value from the domain into the function. For our example, \( f = \{(1,3),(2,6),(3,11),(4,18)\} \), we take the second element from each pair to define our range, which is \( \{3, 6, 11, 18\} \).

Understanding the range helps us realize the outcomes our function can deliver. It answers the question: what values are possible when we transform our inputs through the function's rule? Much like the domain sets the journey's start, the range shows us the possible endpoints. The simplicity of our function made finding the range straightforward, but for more complicated functions, establishing the range can require solving equations or analyzing behavior at limits.

Inverse Functions

Inverse functions are an exciting topic in algebra because they essentially navigate the function's map in reverse. If a function can be thought of as a rule that converts input to output, an inverse function tells us how to get back to the original input from the output. We denote the inverse function by \( f^{-1} \).

To find the inverse, we switch each input-output pair. For \( f = \{(1,3),(2,6),(3,11),(4,18)\} \), the inverse function is \( f^{-1} = \{(3,1),(6,2),(11,3),(18,4)\} \). This new function essentially reverses the roles of domain and range.

• The domain of \( f^{-1} \) is the range of \( f \), which here is \( \{3, 6, 11, 18\} \).
• The range of \( f^{-1} \) is the domain of \( f \), which here is \( \{1, 2, 3, 4\} \).

Inverse functions are handy because they can help solve equations efficiently. By mapping backwards, they allow us to undo operations performed by the original function, making them a powerful mathematical tool.