Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(x), its inverse f⁻¹(x) satisfies the condition that f(f⁻¹(x)) = x for all x in the domain of f⁻¹, and f⁻¹(f(x)) = x for all x in the domain of f. This means that applying the function and then its inverse returns the original input.

Function Composition

Function composition involves combining two functions to create a new function. If you have two functions f and g, the composition f(g(x)) means you apply g first and then apply f to the result. Understanding composition is crucial for verifying the properties of inverse functions, as it demonstrates how they interact with each other.

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the case of f(x) = 2x, it is a linear function with a slope of 2 and no y-intercept. The simplicity of linear functions makes finding their inverses straightforward, as they are one-to-one and can be easily manipulated algebraically.