Composite Function

In mathematics, a composite function is a function that is formed by the composition of two or more functions. It is a way of combining functions to create a new function. The composite function is denoted by the symbol "∘" (read as "composed with").

Definition

Let f: A → B and g: B → C be two functions. The composite function of f and g, denoted as g∘f, is a function from A to C defined by (g∘f)(x) = g(f(x)) for all x in A.

In other words, the composite function g∘f takes an input x, applies the function f to it, and then applies the function g to the result of f(x).

Notation

The composite function g∘f is read as "g composed with f" or "g after f". The order of the functions is important, as g∘f is not the same as f∘g unless f and g are both identity functions.

Example

Let's consider two functions f(x) = 2x and g(x) = x + 3. We can find the composite function g∘f as follows:

(g∘f)(x) = g(f(x)) = g(2x) = 2x + 3

So, the composite function g∘f is given by (g∘f)(x) = 2x + 3.

Properties of Composite Functions

1. Associativity: The composition of functions is associative, which means that for three functions f, g, and h, (h∘g)∘f = h∘(g∘f).

2. Identity Function: The identity function, denoted as I, is a function that returns the same value as its input. For any function f, f∘I = I∘f = f.

3. Domain and Range: The domain of the composite function g∘f is the set of all x in the domain of f such that f(x) is in the domain of g. The range of the composite function g∘f is the set of all y in the range of f such that g(y) is in the range of g.

4. Inverse Functions: If f and g are inverse functions, then g∘f = I and f∘g = I, where I is the identity function.

Examples

1. Let f(x) = 2x and g(x) = x + 3. Find the composite function g∘f.

(g∘f)(x) = g(f(x)) = g(2x) = 2x + 3

So, the composite function g∘f is given by (g∘f)(x) = 2x + 3.

1. Let f(x) = √x and g(x) = x^2. Find the composite function f∘g.

(f∘g)(x) = f(g(x)) = f(x^2) = √(x^2) = |x|

So, the composite function f∘g is given by (f∘g)(x) = |x|.

Table of Differences

Property	Composite Function	Inverse Function
Associativity	(h∘g)∘f = h∘(g∘f)	-
Identity Function	f∘I = I∘f = f	-
Domain and Range	Domain of g∘f = {x in domain of f such that f(x) is in domain of g}	-
Inverse Functions	g∘f = I and f∘g = I	If f and g are inverse functions

Conclusion

Composite functions provide a way to combine multiple functions into a single function. By applying one function after another, we can create new functions with different properties and behaviors. Understanding composite functions is important in various areas of mathematics, such as calculus, algebra, and geometry.