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.
$$
f(x)=3-\frac{1}{2} x
$$

Key Concepts

Graphing Functions with Technology

Using a graphing utility allows for the visualization of functions, making it easier to analyze properties such as being one-to-one. Graphing calculators or software can quickly plot functions and provide visual feedback to apply tests like the horizontal line test effectively.

Horizontal Line Test

The horizontal line test is a graphical technique used to determine if a function is one-to-one. By observing the graph of the function, if any horizontal line intersects the graph more than once, the function fails the test and therefore does not have an inverse function that is itself a function.

One-to-One Functions

A one-to-one function is one in which every distinct input corresponds to a distinct output. This property ensures that no two different inputs produce the same output, which is a necessary condition for a function to have an inverse that is also a function.

Inverse Function

An inverse function reverses the mapping of the original function such that the roles of inputs and outputs are swapped. A function has an inverse function if and only if it is one-to-one, meaning the inverse will satisfy f?¹(f(x)) = x and f(f?¹(x)) = x for all x in their respective domains.

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