Composite and Inverse Functions

A composite function is a combination of two functions, generally denoted as (fg)(x) or f(g(x)). The output of one function acts as the input for the next.
To find the composite function, substitute the second function into the first. For instance, if f(x) = 3x+2 and g(x) = x^2, then f(g(x)) = 3x^2+2.
The inverse function of a given function f(x) is the function that reverses the effect of the original function, denoted as f^-1(x).
The inverse function conveys that if y = f(x) for a certain value of x, then x = f^-1(y) for that particular value of y.
To find an inverse function, exchange x and y in the original function’s equation and then solve for y.
The composite of a function with its inverse yields the original input value, that is, f(f^-1(x)) = x and f^-1(f(x)) = x.
Illustrate both composite and inverse functions using function diagrams.

Remember, when drawing graphs, the graph of the inverse function is a reflection of the original function’s graph in the line y=x.

Key Concepts for Revision

Understand and apply the concept of function composition.
Know the notation for function composition and inverses.
Ability to calculate or derive a composite function given two functions.
Understand how to find an inverse function algebraically.
Draw and interpret diagrams illustrating composite and inverses.
Understand how to see the geometric meaning of function composition and inverses on graphs.