Find inverse functions and relations

Learning Objectives Define and differentiate between an inverse relation and an inverse function. Determine if a function has an inverse that is also a function by using the Horizontal Line Test. Algebraically find the equation of an inverse function for linear, quadratic, radical, and rational functions. Verify if two functions are inverses of each other using the composition of functions. State the domain and range of an inverse function based on the range and domain of the original function. Describe the graphical relationship between a function and its inverse as a reflection across the line y = x. Ever used an 'undo' button or unscrambled a secret code? 🤫 Inverse functions are the mathematical equivalent of reversing a process! This tutorial explores the con...

TermDefinitionExample Inverse RelationA relation formed by interchanging the independent variable (x) and the dependent variable (y) in a given relation. The set of ordered pairs (x, y) becomes (y, x).If the original relation is R = {(1, 5), (2, 7), (3, 9)}, its inverse relation is R⁻¹ = {(5, 1), (7, 2), (9, 3)}. Inverse FunctionAn inverse relation that is also a function. It is denoted by f⁻¹(x). An inverse function exists only if the original function is one-to-one.If f(x) = x + 4, its inverse function is f⁻¹(x) = x - 4, because it 'undoes' the action of adding 4. One-to-One FunctionA function where each element of the range corresponds to exactly one element of the domain. In other words, no two different inputs produce the same output.f(x) = 2x is one-to-one. However, g(x) =...

Algebraic Method for Finding an Inverse Function 1. Replace f(x) with y. \n2. Swap x and y. \n3. Solve the new equation for y. \n4. Replace y with f⁻¹(x). This four-step algorithm is the primary method for finding the equation of an inverse function. Inverse Function Composition Property Two functions f(x) and g(x) are inverses if and only if (f ∘ g)(x) = f(g(x)) = x and (g ∘ f)(x) = g(f(x)) = x. This property is used to formally verify or prove that two functions are inverses of each other. Both compositions must result in x. Domain and Range Relationship Domain of f⁻¹(x) = Range of f(x) \nRange of f⁻¹(x) = Domain of f(x) The inputs of the inverse function are the outputs of the original function, and vice-versa. This is critical for functions with restricted domain...