Chapter 1
Linear Equations and Graphing
Course: Gre
Subject: Gre Quantitative Reasoning
Chapter: Quadratic Equations And Polynomials
Introduction: Functions Inside Functions
Functions can be combined in various ways. One of the most powerful combinations is function composition, where the output of one function becomes the input of another. This creates a new function that often reveals relationships not visible in the original functions.
Composite Functions
A composite function is formed when one function is substituted into another.
f∘g)(x)=f(g(x)
Read as "f of g of x."
Order Matters
Function composition is not commutative. In general:
f(g(x))≠g(f(x))
Evaluating Composite Functions
Work from the inside out: evaluate the inner function first, then the outer function.
Example
Given f(x)=3xand g(x)=x+4, find f(g(2)).
Step 1: Evaluate g(2)=2+4=6
Step 2: Evaluate f(6)=3×6=18
Result: f(g(2))=18
Order Matters Demonstration
For the same functions at x=0:
- f(g(0))=f(4)=12
- g(f(0))=g(0)=4
Thus, f(g(0))≠g(f(0)).
Domain and Range of Composite Functions