The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f^-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (x +4)/(x-2)

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output is produced by exactly one input. This property ensures that the function has an inverse, as it guarantees that no two different inputs will map to the same output. To determine if a function is one-to-one, you can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes y and returns the original input x. To find the inverse, you typically swap the roles of x and y in the equation and solve for y. This process is crucial for verifying the correctness of the inverse by checking that f(f^-1(x)) = x and f^-1(f(x)) = x.

Verification of Inverse Functions

Verifying that two functions are inverses involves demonstrating that applying one function to the result of the other returns the original input. This is done through two equations: f(f^-1(x)) = x and f^-1(f(x)) = x. If both equations hold true, it confirms that the functions are indeed inverses of each other, ensuring that the operations of the functions and their inverses are consistent and correctly defined.