Identify inverse functions

Learning Objectives Define an inverse function and its relationship to the original function. Verify if two given functions, f(x) and g(x), are inverses of each other using composition. Determine if a function has an inverse that is also a function by applying the Horizontal Line Test to its graph. Identify the domain and range of an inverse function based on the range and domain of the original function. Recognize the graphical relationship between a function and its inverse as a reflection across the line y=x. Apply the concept of inverse functions to common function pairs, such as logarithmic and exponential functions. Ever sent a secret message that needed a key to decode? 🤫 Inverse functions work just like that, providing the 'key' to undo the original operat...

TermDefinitionExample Inverse FunctionA function that reverses or 'undoes' the action of another function. If a function f takes an input x to an output y, its inverse, denoted f⁻¹(x), takes the output y back to the original input x.If f(x) = x + 5, then f(2) = 7. The inverse function is f⁻¹(x) = x - 5, and f⁻¹(7) = 2. One-to-One FunctionA function where every distinct input value corresponds to a distinct output value. No two different inputs produce the same output. Only one-to-one functions have inverse functions.f(x) = 2x is one-to-one. However, f(x) = x² is not one-to-one because f(2) = 4 and f(-2) = 4. Horizontal Line TestA graphical method to determine if a function is one-to-one. If any horizontal line can be drawn that intersects the graph of the function at more than o...

Inverse Function Composition Rule Two functions f(x) and g(x) are inverses if and only if both conditions are met: (f ∘ g)(x) = f(g(x)) = x AND (g ∘ f)(x) = g(f(x)) = x. This is the primary algebraic test to confirm if two functions are inverses. You must check the composition in both directions. Inverse Function Notation The inverse of a function f(x) is denoted as f⁻¹(x). It is critical to understand that the '-1' is not an exponent. f⁻¹(x) does NOT mean 1/f(x) (the reciprocal). Graphical Reflection Principle The graph of f⁻¹(x) is the reflection of the graph of f(x) across the line y = x. This rule allows you to visualize the inverse of a function. If you have a point (a, b) on the graph of f(x), then the point (b, a) must be on the graph of f⁻¹(x).