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A one-to-one function is a special type of function with a unique property: each element in the domain (input) maps to a unique element in the range (output). To put it simply, if you have two different inputs, they will never produce the same output.
To identify a one-to-one function, you first need to understand this unique mapping. When you graph a one-to-one function, no horizontal line will intersect the graph more than once. This is known as the horizontal line test, which we will explain in more detail below.
Determining whether a function is one-to-one is crucial because only one-to-one functions have inverses that are also functions.
So, one-to-one functions ensure that each input is paired with a unique output, and it guarantees the availability of an inverse function.

The horizontal line test is a simple visual method to check if a function is one-to-one. Here's how it works:

• Draw or imagine horizontal lines across the graph of the function.
• If any horizontal line intersects the graph at more than one point, the function is not one-to-one.
• If all horizontal lines intersect the graph at exactly one point or less, the function is one-to-one.

Let's apply this to the function given in the exercise: \( f(x)=-\frac{1}{2} x-2 \). This function is a straight line with a negative slope. Think about the graph of any linear function. It looks like a straight line that's either increasing or decreasing. When you draw horizontal lines, they will intersect our linear graph at only one point because it's continuously moving in a single direction (either up or down).
Thus, by using the horizontal line test, we can confirm that our given function is indeed one-to-one.

Linear functions are functions of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In other words, they represent straight lines when graphed.
In our exercise, the given function is \( f(x) = -\frac{1}{2}x - 2 \). Here, \( m = -\frac{1}{2} \) and \( b = -2 \). The slope of a linear function, which in this case is \(-\frac{1}{2}\), determines the steepness and direction of the line. The y-intercept is the point where the line crosses the y-axis.
When the slope \( m \) is non-zero, the line will always pass the horizontal line test as explained above, ensuring that the function is one-to-one. This is essential because only one-to-one functions have inverses that are also functions.
Now that we know our function is linear and one-to-one, we can confidently find its inverse by swapping the roles of \( x \) and \( y \) and solving for \( y \). This leads us to finding the inverse function, \( f^{-1}(x) = -2x - 4 \).