In Exercises 35-46, use a graphing utility to graph the...

In Exercises $35-46,$ use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function exists.
$$
h(x)=|x+4|-|x-4|
$$

Key Concepts

Inverse Function

An inverse function reverses the inputs and outputs of the original function, effectively 'undoing' the function's operation. For a function to have an inverse that is also a function, it must be one-to-one, which can often be verified using the Horizontal Line Test.

One-to-One Function

A one-to-one function, or injective function, is one where each output corresponds to exactly one input. Establishing that a function is one-to-one is essential for determining the existence of an inverse function, since only one-to-one functions can be inverted.

Horizontal Line Test

The Horizontal Line Test is a graphical method used to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, the function fails the test and does not have an inverse that is also a function.

Absolute Value Function

An absolute value function outputs the non-negative magnitude of a number, resulting in a V-shaped graph. It is fundamental in understanding how distances are measured without regard to direction, and is often used in forming piecewise definitions of functions.

Piecewise Function

A piecewise function is defined by different expressions over different intervals of the domain. This is common when absolute value functions are involved, as their definition inherently splits the function's behavior based on the sign of the expression inside the absolute value.

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