Problem 6 - 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. \(f=\\{(-1,-1),(1,1),(0,2),(2,0)\\}\)

Step-by-Step Solution

Answer

The function is one-to-one, and its inverse is \(f^{-1} = \{(-1, -1), (1, 1), (2, 0), (0, 2)\}\).

Step 1: Understanding One-to-One Functions

A one-to-one function, also known as an injective function, is a function for which every element in the domain corresponds to a unique element in the range. In other words, no two different elements in the domain of the function should map to the same element in the co-domain. Thus, we need to check that each input (the first number in each pair) maps to a unique output (the second number in each pair).

Step 2: Check for Unique Outputs

Examine the given function: \( f = \{(-1,-1),(1,1),(0,2),(2,0)\}\)- For \(-1\), the output is \(-1\).- For \(1\), the output is \(1\).- For \(0\), the output is \(2\).- For \(2\), the output is \(0\).Since no two different inputs produce the same output, the function is one-to-one.

Step 3: Switch Inputs and Outputs

To find the inverse function, switch each pair’s coordinates (input and output). For the set \(f\):- \((-1, -1)\) becomes \((-1, -1)\).- \((1, 1)\) becomes \((1, 1)\).- \((0, 2)\) becomes \((2, 0)\).- \((2, 0)\) becomes \((0, 2)\).Thus, the inverse function is given by the set \(f^{-1} = \{(-1, -1), (1, 1), (2, 0), (0, 2)\}\).

Step 4: Validate the Inverse Function

To ensure correctness, check that after swapping, each new input still maps to a unique new output. Each coordinate pair in the set \(f^{-1}\) does not repeat any output for different inputs, confirming it remains a function.

Key Concepts

Inverse Function

An inverse function is like a reverse gear for mathematical functions. It allows you to go backward from the output back to the input. Imagine walking down a path using a map, and then retracing your steps exactly using the same path every time.
In mathematical terms, if you have a function \( f(x) \), its inverse is denoted as \( f^{-1}(x) \).
They have this special relationship: when you apply \( f \) and then \( f^{-1} \), you should end up where you started. Mathematically, this is written as:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

Finding an inverse involves switching each input and output pair. For functions built from coordinate pairs, like the example \( f=\{(-1,-1),(1,1),(0,2),(2,0)\} \), reversing involves swapping these pairs to get the inverse function.
This gives us new sets of pairs, forming what we call the inverse function. Each input now becomes an output, forming a sort of mirror image of the original function.

Unique Output

The concept of unique output is a cornerstone of understanding one-to-one or injective functions.
In any function, you should think of it as a matching activity. Each input from the domain must correspond to just one output in the range. If you feed in a specific input, you should always get back the same output.
This is what it means for an output to be unique:

• Different inputs should never lead to the same output
• Each pair of input-output is distinct and doesn't repeat the output for different inputs

For example, taking the function \( f = \{(-1,-1), (1,1), (0,2), (2,0)\} \), let's ensure that each input yields a different output:

• For input \(-1\), the output is \(-1\)
• For input \(1\), the output is \(1\)
• For input \(0\), the output is \(2\)
• For input \(2\), the output is \(0\)

Because all outputs are unique, this function qualifies as one-to-one, meaning it’s safe for reverse journeys through an inverse function without trouble.

Injective Function

Injective functions are a major category of functions, and are sometimes referred to as one-to-one functions.
These functions have the property that no two different inputs in their domain produce the same output in their codomain. Thus, injective functions are all about ensuring every input has an individual and unique output.
Thinking about injectivity is quite like thinking about putting keys into locks. Each key (input) opens exactly one lock (output), and each lock is only opened by one key.

• An injective function will never "recycle" its output for different inputs.
• This ensures the possibility of finding a valid inverse function.

For example, examining \( f=\{(-1,-1),(1,1),(0,2),(2,0)\} \), you can see that:

• \(-1\) maps uniquely to \(-1\)
• \(1\) maps uniquely to \(1\)
• \(0\) maps uniquely to \(2\)
• \(2\) maps uniquely to \(0\)

Each input having a unique output means this function is injective, confirming it can have an inverse just by swapping the pairs around. This specificity and uniqueness are what make injective functions so powerful in mathematical operations.