Definition of Composite of Two Functions: The composition of the functions f and
g is given by
(f o g)(x) = f(g(x)).
A composite function can be viewed as a function within a function, where the
composition
(f o g)(x) = f(g(x)).
has f as the "outer" function and g as the "inner" function. This is reversed in the
composition
(g o f)(x) = g(f(x)).
The domain of the composite function (f o g) is the set of all x in the domain of g such that g(x) is in the domain of f.
Example #1: Comparing the Compositions of Functions
A function f is one-to-one if each value of the dependent variable (the second term in an ordered pair) corresponds to exactly one value of the independent variable (first term in an ordered pair).
By interchanging the first and second coordinates of an ordered pair, you can form another function called the inverse function of f, denoted by f-1.
Only one-to-one functions have inverses that are functions.
Horizontal Line Test for Inverse Functions
A function f has an inverse function f-1 if and only if no horizontal line intersects the graph of f at more than one point.
Example #2: Use the Horizontal Line Test to determine if the function is one-to-one and so has an inverse function.
a) g(x) = |x - 4|
b) f(x) = √-x
Inverse Function
Only one-to-one functions have inverses that are functions.
Finding Inverses given an Equation Algebraically:
How to find f-1
Rewrite f(x) as y
Exchange x and y
Solve for y
Rewrite y as f-1(x)
Verify that f and f-1 are inverse functions of each other by showing that f(f-1(x)) = x = f-1(f(x)).
Example #3: Find the inverse of each function and graph (blue = original and black inverse):
Notice that the graphs of inverses share a unique relationship. They are mirror images of each other across the line y = x. Therefore, you can verify that functions are inverses by graphing them.