Identify inverse functions

Learning Objectives Algebraically verify if two functions are inverses using the composition property. Apply the Horizontal Line Test to determine if a function has an inverse that is also a function. Identify the graph of an inverse function as a reflection across the line y = x. Determine the inverse of a function represented by a set of ordered pairs. State the domain and range of an inverse function given the domain and range of the original function. Differentiate between the notation for an inverse function, f⁻¹(x), and the reciprocal, 1/f(x). How does a secure website encrypt and then decrypt your password? It uses a function and its perfect 'undo' function! 🔐 This tutorial focuses on the methods used to identify and verify inverse functions. You will lear...

TermDefinitionExample Inverse FunctionA function, denoted as f⁻¹(x), that reverses the action of another function, f(x). If f(a) = b, then f⁻¹(b) = a.If f(x) = x + 5, its inverse is f⁻¹(x) = x - 5. Applying f to 3 gives 8, and applying f⁻¹ to 8 gives 3. One-to-One FunctionA function where each input value (x) corresponds to a unique output value (y), and each output value corresponds to a unique input value. Only one-to-one functions have inverse functions.f(x) = x³ is one-to-one. f(x) = x² is not, because f(2) = 4 and f(-2) = 4 (one output from two inputs). Horizontal Line TestA graphical test to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, it is not one-to-one and does not have an inverse that is a function.A horizont...

Algebraic Test for Inverses (Composition) Two functions, f(x) and g(x), are inverses of each other if and only if (f \circ g)(x) = f(g(x)) = x and (g \circ f)(x) = g(f(x)) = x for all x in the respective domains. This is the definitive algebraic method to prove two functions are inverses. You must test the composition in both directions. Graphical Test for Inverses (Reflection) The graph of y = f⁻¹(x) is the reflection of the graph of y = f(x) across the line y = x. Use this property to visually inspect two graphs to see if they could be inverses or to sketch the graph of an inverse. Domain and Range Relationship Domain(f) = Range(f⁻¹) and Range(f) = Domain(f⁻¹) This rule allows you to determine the domain and range of an inverse function without having to find the i...