Problem 5 - Algebra A Combined Function

Question

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. \(f=\\{(11,12),(4,3),(3,4),(6,6)\\}\)

Step-by-Step Solution

Answer

The function is one-to-one; inverse is \(f^{-1}=\{(12,11),(3,4),(4,3),(6,6)\}\).

Step 1: Understanding a One-to-One Function

A one-to-one function means that each output value has exactly one unique input value. In other words, there cannot be two different inputs that result in the same output.

Step 2: Analyzing the Given Function

Let's analyze the function \(f=\{(11,12),(4,3),(3,4),(6,6)\}\). Check to see if any output value is repeated for different input values.

Step 3: Checking for Repeated Outputs

Look at the outputs: 12, 3, 4, and 6. None of these outputs repeat for different inputs, meaning each output corresponds uniquely to one input. This means the function \(f\) is one-to-one.

Step 4: Finding the Inverse Function

Since we confirmed that \(f\) is one-to-one, we can find its inverse by switching the inputs and outputs. The inverse will be \(f^{-1}=\{(12,11),(3,4),(4,3),(6,6)\}\).

Step 5: Verifying the Inverse

Check the inverse function to ensure that each input corresponds to a unique output. The inverse \(f^{-1}=\{(12,11),(3,4),(4,3),(6,6)\}\) is also a function, confirming it is the inverse of \(f\).

Key Concepts

Inverse Functions

An inverse function essentially undoes what the original function does. The idea is similar to reversing a process: you start from the result and go back to the starting point. For a function to have an inverse, it must be a one-to-one function. This means that each output has a unique input. If you take a coordinate pair

• (x, y)

in the original function, reversing it to (y, x) should also form a valid function. The inverse allows us to find these unique input values by "swapping" the roles of inputs and outputs. Thus, the inverse of a function \( f \) is denoted by \( f^{-1} \), not to be mistaken with simple division or arithmetic inverse. It's crucial to ensure the original function is one-to-one to define this inverse properly.

Function Analysis

Function analysis involves breaking down the characteristics of a function to understand its behavior. One of the first things to check is whether the function is one-to-one. This helps ascertain if an inverse function exists. In our analysis of function \( f = \{(11,12),(4,3),(3,4),(6,6)\} \), we noted that each output corresponds to a unique input.

• Output 12 has input 11.
• Output 3 has input 4.
• Output 4 has input 3.
• Output 6 has input 6.

Since all outputs are unique, there is no input with more than one output, confirming it is indeed one-to-one. By performing this analysis, we verify that we can proceed to find its inverse. In summary, function analysis not only confirms the presence of an inverse but also helps identify any patterns or properties of the function.

Coordinate Systems

A coordinate system allows us to describe and plot the location of points on a graph. In the context of functions and their inverses, coordinate pairs are essential. For instance, the given function \( f \) is presented as a set of ordered pairs within a coordinate system. Each (x, y) pair represents a useful snapshot of the input-output relationship.
When finding an inverse, we swap these coordinates to \( (y, x) \), which retains the relationship but inverts the role of each variable.

• Original pair: (11, 12)
• Inverse pair: (12, 11)

In this way, the coordinate system provides a simple, visual method for working with and understanding functions and their inverses. It emphasizes the symmetry that exists around the line \( y = x \) for a true inverse, where the function and its inverse are mirror images of each other.