Find values of inverse functions from graphs

Learning Objectives Define the relationship between a function and its inverse graphically. Identify the coordinates of a point on the graph of an inverse function, f⁻¹(x), given the graph of the original function, f(x). Evaluate f⁻¹(a) for a specific value 'a' by interpreting the graph of f(x). Explain the graphical relationship between the domain and range of a function and its inverse. Determine if a function has an inverse that is also a function by applying the Horizontal Line Test to its graph. Solve composite function problems involving inverses, such as f(f⁻¹(b)) or f⁻¹(f(a)), using a graph. Ever wondered how to 'undo' a function just by looking at its picture? 🖼️ Let's flip our perspective and learn how to read a graph in reverse! This tuto...

TermDefinitionExample Inverse Function (f⁻¹(x))A function that 'reverses' or 'undoes' another function. If the original function f takes an input 'a' to an output 'b', the inverse function f⁻¹ takes the input 'b' back to the output 'a'.If f(x) = x³ and we know f(2) = 8, then the inverse function f⁻¹(x) = ³√x gives f⁻¹(8) = 2. One-to-One FunctionA function where each output value (y) is associated with exactly one input value (x). Only one-to-one functions have inverse functions.f(x) = 2x + 1 is one-to-one. f(x) = x² is not, because f(2) = 4 and f(-2) = 4 (the output 4 comes from two different inputs). Horizontal Line TestA visual method to determine if a function is one-to-one. If any horizontal line can be drawn that intersects...

The Fundamental Inverse Relationship If f(a) = b, then f⁻¹(b) = a. This is the core principle for finding inverse values from a graph. To find f⁻¹(b), you are looking for the x-value on the graph of f(x) that produces the y-value of b. Coordinate Swap Rule (a, b) on f(x) \iff (b, a) on f⁻¹(x) This rule translates the algebraic relationship into a graphical one. A point on the original function's graph has its coordinates swapped to become a point on the inverse function's graph. Composition Cancellation Properties f(f⁻¹(x)) = x and f⁻¹(f(x)) = x These rules show that a function and its inverse 'cancel' each other out. This holds for all x in the domain of the inner function. Graphically, this means finding a value, applying f, and then applying...