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In mathematics, a function is called one-to-one if each output value is associated with exactly one unique input value. This means that no two different inputs produce the same output. This property is crucial when determining if a function has an inverse because only one-to-one functions are guaranteed to have inverses.
To identify a one-to-one function, we can use various criteria:

• An algebraic test: Check if the function passes the one-to-one criteria algebraically, meaning if \(f(a) = f(b)\) implies \(a = b\).
• A graphical test: Use the Horizontal Line Test, where a graph of the function that is intersected by any horizontal line at most once indicates a one-to-one function.

For the function \(h(x) = -10 - x\), it passes the Horizontal Line Test because it is a straight line with a slope of -1, which is strictly decreasing. Hence, this function is one-to-one.

The Horizontal Line Test is a graphical method to determine if a function is one-to-one. If any horizontal line crosses the graph of a function at most once, then the function is one-to-one. This test helps to visually verify the one-to-one property without delving into complex algebra.
For instance, consider the graph of \(h(x) = -10 - x\). This is a straight line with a negative slope, which continuously decreases. A horizontal line drawn at any level will intersect the graph at exactly one point, confirming that \(h(x)\) is one-to-one.
The Horizontal Line Test is particularly useful because it provides a quick visual check and is especially handy when dealing with complex functions. However, for accurate results, it's essential to understand the behavior of the function fully.

Function composition involves combining two functions where the output of one function becomes the input of another. For a function \(h\) and its inverse \(h^{-1}\), function composition is used to verify if \(h^{-1}\) is indeed the correct inverse.
When verifying inverses, we perform the following operations:

• Compose \(h\) with \(h^{-1}\) and check if the result is the identity function. For the given example, \(h(h^{-1}(x)) = h(-x - 10)\). Simplify: \(-10 - (-x - 10) = x\).
• Next, compose \(h^{-1}\) with \(h\) to ensure it also gives the identity function: \(h^{-1}(h(x)) = h^{-1}(-10 - x)\). Simplify: \(-(-10 - x) - 10 = x\).

If both compositions return the input \(x\), then the inverse has been correctly found. In this case, both operations return \(x\), thus \(h^{-1}(x) = -x - 10\) is indeed the correct inverse of \(h(x) = -10 - x\).