Lesson 8 of 10 · 8 min
Composition of functions
NCERT §1.4
After the 100 m final, each runner gets a finishing position, and each position gets house points: 5 for first, 3 for second, 1 for third, 0 for fourth. The scoreboard goes straight from runner to points.
The lesson in notes
In short
For f : A → B and g : B → C, the composition gof : A → C is gof(x) = g(f(x)) for every x ∈ A. f acts first, then g.
gof is defined only when every output of f is an allowed input of g, that is, when the range of f lies inside the domain of g.
Finite example: f(1) = 4, f(2) = 5, f(3) = 5 and g(4) = 7, g(5) = 9 give gof(1) = g(4) = 7, gof(2) = g(5) = 9 and gof(3) = g(5) = 9.
Order matters. For f(x) = x + 2 and g(x) = x² on R, gof(x) = (x + 2)² and fog(x) = x² + 2; at x = 1 these are 9 and 3, so gof ≠ fog.
Read gof from the right: the letter next to x acts first. gof(x) = g(f(x)) and fog(x) = f(g(x)).
Composing with the identity changes nothing: foI_X = f and I_Yof = f for f : X → Y.