Problem 43 - A Graphical Approach to College Algebra

Question

Determine whether each function is one-to-one. If so, find its inverse. $$f=\\{(10,4),(20,5),(30,6),(40,7)\\}$$

Step-by-Step Solution

Answer

Function is one-to-one; inverse is \( f^{-1} = \{(4,10),(5,20),(6,30),(7,40)\} \).

Step 1: Understanding One-to-One Functions

A function is considered one-to-one (injective) if each element in the domain maps to a unique element in the codomain. This means no two different elements in the domain map to the same element in the codomain.

Step 2: Verify the One-to-One Property

Examine the given function \( f = \{(10,4),(20,5),(30,6),(40,7)\} \). Each input value (10, 20, 30, 40) maps to a unique output value (4, 5, 6, 7), and no two inputs share the same output. Thus, this function is one-to-one.

Step 3: Finding the Inverse of the Function

To find the inverse of a one-to-one function, swap the elements in each pair so that each output becomes an input and vice versa. For \( f \), the inverse \( f^{-1} \) is obtained by swapping each pair: \( \{(4,10),(5,20),(6,30),(7,40)\} \).

Step 4: Check the Validity of the Inverse Function

Verify that the inverse function \( f^{-1} = \{(4,10),(5,20),(6,30),(7,40)\} \) is also a function by ensuring that each input value maps to a single output value. Since each element is unique, \( f^{-1} \) is also a valid function.

Key Concepts

Injective Functions

When talking about injective functions, we refer to functions where each input leads to a unique output. Imagine handing out badges where each person gets exactly one badge and no two people share the same badge number. This uniqueness is key to understanding injective functions. For a function to be one-to-one:

• Each element in the domain (inputs) must map to a distinct element in the codomain (outputs).
• No two different input values share the same output value.

In the problem given, the function is described by pairs like \(10,4\), \(20,5\), \(30,6\), and \(40,7\). Notice that each number from the domain (10, 20, 30, 40) has its unique partner in the codomain (4, 5, 6, 7). Thus, we see it's indeed one-to-one or injective. Understanding this property is crucial if we want to move on to find inverse functions.

Inverse Functions

An inverse function essentially reverses the original function. Imagine if you had a list of pairs and you simply swap each input-output pair's positions. This new list represents the inverse function.To find the inverse of an injective function, follow these steps:

• Swap each input with its respective output in every pair.
• By swapping, the original output becomes the input, and the original input becomes the output.

In our given function \( f = \{(10,4), (20,5), (30,6), (40,7)\} \), swapping the pairs results in the inverse function \( f^{-1} = \{(4,10), (5,20), (6,30), (7,40)\} \).This transforms the mappings so that now, if you provide the number 4, the output is 10, effectively reversing the original roles of concepts in the function. Keep in mind that swapping is possible and valid only when the function is one-to-one.

Function Verification

Verifying a function and its inverse ensures that both truly represent the relationship they claim. This process adds confidence to our work by ensuring the pairs are consistent and legit.Here's what to check when verifying:

• Each input maps to precisely one output – no duplicates in outputs.
• Validate that invertibility is maintained; the inverse must map correctly for each output-input swap.

For instance, the original function \( f = \{(10,4), (20,5), (30,6), (40,7)\} \) was injective, hence its inverse \( f^{-1} = \{(4,10), (5,20), (6,30), (7,40)\} \) should also be checked.Ensure that each new input from the inverse (4, 5, 6, 7) leads to one and only one unique output (10, 20, 30, 40). This illustrates the completeness of functioning both ways, affirming the inverse is valid.