Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = -4x+2, g(x) = -1/4x - 2

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If f(x) is a function, its inverse, denoted as f⁻¹(x), satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f. This means that applying the function and then its inverse returns the original input.

Composition of Functions

The composition of functions involves combining two functions to form a new function. For two functions f and g, the composition is denoted as (f ∘ g)(x) = f(g(x)). To verify if f and g are inverses, we need to check if both f(g(x)) = x and g(f(x)) = x hold true for all x in their respective domains.

Linear Functions

Linear functions are functions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. They graph as straight lines and have constant rates of change. Understanding the properties of linear functions is essential when determining their inverses, as the inverse of a linear function is also linear, provided the slope is not zero.