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Inverse Functions: A Comprehensive Overview

If two functions, f(x) and g(x), are inverse functions of each other, this means they "undo" each other's operations. Formally, this is defined as:

• f(g(x)) = x for all x in the domain of g(x)
• g(f(x)) = x for all x in the domain of f(x)

Key Characteristics & Implications:

• Reflection: The graphs of f(x) and g(x) are reflections of each other across the line y = x.

• Swapping of x and y: To find the inverse of a function, you essentially swap the roles of x and y and solve for y. If y = f(x), then to find the inverse, rewrite this as x = f(y) and solve for y. The solution will be y = g(x), where g(x) is the inverse function.

• One-to-One Functions: A function has an inverse if and only if it is a one-to-one function. This means that each y-value corresponds to exactly one x-value (and vice-versa). The horizontal line test is used to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, it's not one-to-one and does not have an inverse function.

• Notation: The inverse of f(x) is often denoted as f^-1(x). Important: f^-1(x) does not mean 1/f(x).

• Range and Domain Swap: The domain of f(x) becomes the range of f^-1(x), and the range of f(x) becomes the domain of f^-1(x).

• Composition: The composition of a function with its inverse, in either order, results in the identity function (f(f^-1(x)) = x and f^-1(f(x)) = x).