Find the inverse of the given one-to-one function f Give the domain and the range of fand of f^-

step 1 So for this problem we're given the function f of x, which I have graphed over here, has a y int

Find the inverse of the given one-to-one function f Give the...

Key Concepts

One-to-One Functions

A one-to-one function is a function in which every output is produced by exactly one input. This property guarantees that the function has an inverse since each y-value corresponds uniquely to an x-value. In other words, there is no repetition in the output values, ensuring that the process of 'reversing' the mapping is well-defined.

Inverse Functions

The inverse of a function, denoted as f?¹, essentially reverses the roles of the inputs and outputs of the original function. To find an inverse, one typically interchanges the x and y in the function's equation and then solves for y. This new function, if it exists, will 'undo' the operation of the original function, meaning that applying the inverse function to an output will yield the original input.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible outputs. When forming the inverse of a function, these sets are switched; the domain of the inverse function is the range of the original function and vice versa. Identifying these sets is crucial when discussing both the function and its inverse to understand where they are defined.

Graphical Representation of Functions and Inverses

Graphing a function alongside its inverse is a helpful visual tool to illustrate their relationship. The graph of an inverse function is a reflection of the original function’s graph across the line y = x. This reflection property is a key characteristic that helps confirm the correctness of the inverse found, as each point (a, b) on the function corresponds to the point (b, a) on its inverse.

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