COMPOSITION OF FUNCTIONS

Composition Of Functions

Definition Of Composition Of Functions

Composition of Functions is the process of combining two functions where one function is performed first and the result of which is substituted in place of each x in the other function.

More About Composition of Functions

The composition of functions f and g is written as f o g.
[f o g](x) = f[g(x)]
Composition of functions is not commutative. f[g(x)] is generally not equal to g[f(x)].
For example, consider f(x) = 2x and g(x) = x - 3.
f[g(x)] = 2(x - 3) = 2x - 6
g[f(x)] = (2x) - 3 = 2x - 3
f[g(x)] is not equal to g[f(x)].

Video Examples: Function Compositions

Solved Example onComposition of Functions

Ques: Evaluate the composite function f[g(x)] for f(x) = 3x 2 + 6 and g(x) = x - 8.

Choices:

A. x - 8
B. 3x 2 - 48x + 198
C. 3x 2 - 2
D. 3x 2 + 6
Correct Answer: B

Solution:

Step 1: f[g(x)] = f[x - 8]
Step 2: = 3(x - 8)2 + 6
Step 3: = 3(x 2 - 16x + 64) + 6
Step 4: = 3x 2 - 48x + 198.

Quick Summary

Apply the inner function first.
Substitute the result into the outer function.
Composition of functions is generally not commutative: f[g(x)] ≠ g[f(x)].

\[ [f \circ g](x) = f[g(x)] \]

🍎 Teacher Insights

Emphasize that composition means plugging one function into another, not multiplying them. Use visual aids like diagrams to illustrate the process. Provide plenty of practice problems with varying levels of difficulty.

🎓 Prerequisites

Functions
Algebra
Order of Operations

Check Your Knowledge

Q1: If f(x) = x + 2 and g(x) = 3x, what is f[g(x)]?

A. 3x + 2 B. 3x + 6 C. x + 5 D. 3(x + 2)

Frequently Asked Questions

Q: Is f[g(x)] always different from g[f(x)]?
A: Generally yes, but there can be cases where f[g(x)] = g[f(x)] for specific functions and values of x.