91Ó°ÊÓ

Understanding one-to-one functions is crucial for grasping the concept of inverse functions. A function is called one-to-one (or injective) if and only if it never assigns the same value to two different inputs.
This means, for any two different inputs, say, \(x_1\) and \(x_2\), the outputs \(f(x_1)\) and \(f(x_2)\) are always different. Mathematically, we write this condition as: If \(x_1 ≠ x_2\), then \(f(x_1) ≠ f(x_2)\), and vice versa.
Here are a few important points to make this concept more digestible:

• A one-to-one function passes the horizontal line test. No horizontal line intersects the graph of the function more than once.
• This uniqueness property is what allows one-to-one functions to have inverse functions.
• If a function is not one-to-one, it cannot have an inverse.

Imagine a function that assigns each student in a class to a unique locker number. No two students share the same locker. That’s a straightforward example of a one-to-one function.

The graph of an inverse function is a reflection of the graph of the original function over the line \(y = x\). This line acts like a mirror. To visualize this, let's break it down into simple steps:

• Start by plotting the original function \(f\) on a graph.
• Next, imagine the line \(y = x\), which runs diagonally from the bottom-left to the top-right of the graph. This is our mirror line.
• To get the graph of the inverse function \(f^{-1}\), reflect each point of the original function across this mirror line.

For instance, if the point \((2, 3)\) is on the graph of \(f\), the point \((3, 2)\) will be on the graph of \(f^{-1}\). Notice how the coordinates swap places!
Don't forget these key points:

• The graph of the inverse function will intersect the graph of the original function wherever they both touch the line \(y = x\).
• If \(f\) is one-to-one and its graph passes the horizontal line test, its inverse will be a function too.

This flipping of coordinates from \((a, b)\) to \((b, a)\) ensures that the original and inverse functions are exact opposites.

When working with functions and their inverses, understanding how to handle coordinates is essential.
For any function \(f\), a point \((a, b)\) on its graph tells us that \(f(a) = b\). Here's how this translates to the inverse function:

• The point \((a, b)\) on the graph of \(f\) will become the point \((b, a)\) on the graph of \(f^{-1}\).
• This coordinate swap happens because the inverse function essentially reverses the input and output of the original function.

Let's break it down further:

• If you know a point \((a, b)\) is on the function's graph, then by definition, \(f(a) = b\).
• For the inverse function, \(f^{-1}(b) = a\), because it's undoing the original function's calculation.
• This means that if you have a point \((2, 5)\) on \(f\), you'll have \((5, 2)\) on \(f^{-1}\).

Knowing how to swap coordinates correctly between a function and its inverse is fundamental. It helps in plotting these functions and solving problems related to inverses.