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.
$$
f(x)=x^{5}-7
$$

Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, is one where each element of the function’s domain maps to a unique element in the codomain. That means no two different inputs produce the same output. This property is crucial when determining if an inverse function can exist, as only one-to-one functions have well-defined inverses.

Horizontal Line Test

The Horizontal Line Test is a graphical method used to determine if a function is one-to-one. By drawing horizontal lines across the graph, one checks if any line intersects the graph at more than one point. If it does, the function is not one-to-one; if every horizontal line overlaps the graph at most once, the function passes the test, confirming its injectivity.

Inverse Function

An inverse function essentially reverses the operation of the original function, swapping the roles of the input and output. For a function to have an inverse, it must be one-to-one, ensuring that every output is uniquely paired with an input. When an inverse exists, applying the function and then its inverse (or vice versa) will return the original value.

Graphing Utility

A graphing utility is a digital tool or calculator that produces a visual representation of functions. It assists in analyzing the behavior of functions, such as checking for one-to-one properties using the Horizontal Line Test, by presenting an accurate graph that helps in understanding the function's overall behavior.

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