Show that the given function is one-to-one and find its...

Show that the given function is one-to-one and find its inverse. Check your answers algebraically and graphically. Verify that the range of $f$ is the domain of $f^{-1}$ and vice-versa.
$$f(x)=6 x-2$$

Key Concepts

One-to-One Functions

A one-to-one function is one where each output is produced by exactly one input. This means that if f(a) = f(b) then a must equal b. Showing that a function is one-to-one is crucial because only one-to-one functions have well-defined inverses. This concept is central in ensuring that a function's mapping can be reversed uniquely.

Inverse Functions

An inverse function essentially reverses the input-output relationship of the original function. For a function f, its inverse f?¹ satisfies the conditions f(f?¹(x)) = x and f?¹(f(x)) = x. The existence of an inverse function relies on the original function being one-to-one, making the process of inversion an important topic in function analysis.

Algebraic Process for Finding Inverses

Finding an inverse function algebraically typically involves writing the function in terms of y = f(x), then swapping the roles of x and y and solving for y. This method allows us to derive a formula for the inverse function, reinforcing the conceptual understanding that the operations of a function and its inverse undo each other.

Graphical Verification

Graphical verification of a function and its inverse involves plotting both functions and demonstrating that they are reflections of each other across the line y = x. This symmetry provides a visual confirmation that the algebraically derived inverse correctly reverses the input-output relationship of the original function.

Domain and Range Relationships

The domain of a function is the set of all possible inputs, while the range is the set of all outputs. For inverse functions, the range of the original function becomes the domain of the inverse, and vice versa. Understanding this reciprocal relationship is essential in verifying that the inverse function is properly defined over the correct set of values.

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