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A function is called one-to-one (or injective) if every output is produced by exactly one input. This means that no two different inputs should result in the same output. Let’s make it simpler: if you plug in two different values for x, you should get two different values for y. Here's a tip: to determine if a function is one-to-one, you can use the Horizontal Line Test.

To understand this better, consider the linear function in the exercise, $$f(x) = 2x -1$$. Since it is a linear function with a non-zero slope, it will not intersect any horizontal line more than once. This indicates that the function is indeed one-to-one. Isn't that great? You can always rely on this simple test to check for one-to-one functions.

The Horizontal Line Test is a simple, visual way to determine if a function is one-to-one. To perform this test, you need to imagine (or actually draw) horizontal lines across the graph of a function. If any horizontal line crosses the graph more than once, the function fails the test and is not one-to-one.

But if every horizontal line crosses the graph at most once, the function passes the test and is one-to-one.

Let’s check the given function, $$f(x) = 2x - 1$$, one more time. A horizontal line placed on the graph will only cross it once because it’s a linear function with a non-zero slope. Therefore, this function passes the Horizontal Line Test. This technique is a handy tool for quickly checking if a function is one-to-one.

Linear functions are among the simplest types of functions you'll encounter. They have the general form $$f(x) = mx + b$$, where \(m\) is the slope and \(b\) is the y-intercept. These functions graph as straight lines.

For example, in the exercise, we have the function $$f(x) = 2x - 1$$. Here, $$m = 2$$ and $$b = -1$$. Because linear functions with non-zero slopes always have a constant rate of change, they are inherently one-to-one.

This means we can always find an inverse for them. To find the inverse, you replace \(f(x)\) with \(y\), swap \(x\) and \|y\|, and then solve for \(y\). By doing this, you reverse the direction of the function. Following these steps, we can see that the inverse of $$f(x) = 2x - 1$$ is \(f^{-1}(x) = \frac{x + 1}{2} \). Linear functions make it easy to understand and find inverses!