MathematicsChapter 6: Functions and Graphs
Composite Functions
fg(x) means f(g(x)): do g first, then f.
Learning objectives
- Explain composite functions using the correct definitions, conditions, and notation.
- Apply inside first, order matters correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
fg(x) means f(g(x)): do g first, then f. g(f(x)) is the other order and is usually different. f(f(x)) is sometimes written f^2(x) in this course (not the square of the number f(x) unless they say so - read the notation).
Where / symbols
- COMPOSITE FUNCTIONS
- fg(x) means f(g(x)): do g first, then f. g(f(x)) is the other order and is usually different. f(f(x)) is sometimes written f^2(x) in this course (not the square of the number f(x) unless they say so - read the notation).
- TWO FUNCTIONS
- f(x)=x+1, g(x)=x^2. f(g(2))=f(4)=5. g(f(2))=g(3)=9. Write the chain in numbers so you do not swap.
- SAME FUNCTION TWICE
- If f(x)=(1/2)x then f(f(x))=(1/4)x. Inverse undoes: f(f^{-1}(x))=x when the inverse exists.
Overview
Inside, then outside:
Two Functions
f(x)=x+1, g(x)=x^2. f(g(2))=f(4)=5. g(f(2))=g(3)=9. Write the chain in numbers so you do not swap.
Examples f(g(2))=5 · g(f(2))=9 · Not the same
Same Function Twice
If f(x)=(12)x then f(f(x))=(14)x. Inverse undoes: f(f-1(x))=x when the inverse exists.
Examples f(f(x))=x/4 · Inverse undoes · Not [f(x)]^2 unless told
WHAT — COMPOSITE FUNCTIONS
fg(x) means f(g(x)): do g first, then f. g(f(x)) is the other order and is usually different. f(f(x)) is sometimes written f^2(x) in this course (not the square of the number f(x) unless they say so - read the notation). The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Composite Functions matters because functions describe how one quantity depends on another. Domain, range, composition, and inverse ideas are the language used for graphs and mathematical models.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through TWO FUNCTIONS, then SAME FUNCTION TWICE. Worked: f(x) = x + 1, g(x) = x^2. Find f(g(2)) and g(f(2)). g(2) = 2^2 = 4 f(g(2)) = f(4) = 4 + 1 = 5 f(2) = 2 + 1 = 3 g(f(2)) = g(3) = 3^2 = 9 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- f(x) = x + 1, g(x) = x2. Find f(g(2)) and g(f(2)).
- g(2) = 22 = 4
- f(g(2)) = f(4) = 4 + 1 = 5
- f(2) = 2 + 1 = 3
- g(f(2)) = g(3) = 32 = 9
In simple words
fg(x) is not f(x)*g(x). Juxtaposition here means compose, not multiply, unless the chapter says product.
Key points
- inside first
- order matters
- g then f for f(g(x))
- compose with itself
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Composite Functions
Sticky Notes · Quick Revision
Core Idea
- fg(x) means f(g(x)): do g first, then f. g(f(x)) is the other order and is usually different. f(f(x)) is sometimes written f^2(x) in this course (not the square of the number f(x) unless they say so - read the notation).
TWO FUNCTIONS
- f(x)=x+1, g(x)=x^2. f(g(2))=f(4)=5. g(f(2))=g(3)=9. Write the chain in numbers so you do not swap.
SAME FUNCTION TWICE
- If f(x)=(12)x then f(f(x))=(14)x. Inverse undoes: f(f-1(x))=x when the inverse exists.
Worked Method
- Worked: f(x) = x + 1, g(x) = x2. Find f(g(2)) and g(f(2)).
- g(2) = 22 = 4
- f(g(2)) = f(4) = 4 + 1 = 5
- f(2) = 2 + 1 = 3
- g(f(2)) = g(3) = 32 = 9
Common Misconception
- fg(x) is not f(x)*g(x). Juxtaposition here means compose, not multiply, unless the chapter says product.
FBISE Exam Check
- fg(x) is not f(x)*g(x). Juxtaposition here means compose, not multiply, unless the chapter says product.
60-second revision
- fg(x) means f(g(x)): do g first, then f.
- TWO FUNCTIONS: f(x)=x+1, g(x)=x2.
- SAME FUNCTION TWICE: If f(x)=(12)x then f(f(x))=(14)x.
- Worked-method reminder: g(f(2)) = g(3) = 3^2 = 9
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