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+2)³

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 means that if f(a) = f(b), then a must equal b. This property is crucial for finding an inverse function, as only one-to-one functions have inverses that are also functions.

Inverse Functions

An inverse function 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 produces the original input x. To find the inverse, we typically solve the equation y = f(x) for x in terms of y.

Verification of Inverses

To verify that two functions are inverses, we must show that applying one function to the result of the other returns the original input. This is done by demonstrating that f(f^-1(x)) = x and f^-1(f(x)) = x for all x in the domain. This step ensures that the inverse function is correctly derived.