Composite function: fg(x)=f(g(x)).
Read right-to-left like onion layers — the inner one is applied first.
Method (algebraic):
- Find g(x) as an expression.
- Substitute that into f wherever x appears.
- Simplify.
Example. f(x)=2x+3, g(x)=x2−1.
(a) fg(x)=f(g(x))=f(x2−1)=2(x2−1)+3=2x2+1.
(b) gf(x)=g(f(x))=g(2x+3)=(2x+3)2−1=4x2+12x+9−1=4x2+12x+8.
Numerical composite. Easier — work inside-out.
fg(4) with same f,g:
- g(4)=16−1=15.
- f(15)=2(15)+3=33.
Worked qualitative. Why does fg(x)=gf(x) in general?
- Order matters. 'Take square then add 1' is NOT the same as 'add 1 then square'.
- Example: f(x)=x2,g(x)=x+1.
- fg(x)=(x+1)2=x2+2x+1.
- gf(x)=x2+1.
- Different.
Edexcel tip. Mark schemes give M1 for correct order, M1 for substitution, A1 for simplified answer.