Composition: machines in a row
Composition feeds the output of one function straight into another. We write (f ∘ g)(x) = f(g(x)), read “f of g of x.” Work inside out: first run g on x, then run f on whatever g handed back. Order matters — f(g(x)) and g(f(x)) are usually different.
f(x) = x^2 + 1 g(x) = 2x - 3
(f o g)(x) = f(g(x)) = f(2x - 3)
= (2x - 3)^2 + 1
= 4x^2 - 12x + 9 + 1 = 4x^2 - 12x + 10
(g o f)(x) = g(f(x)) = g(x^2 + 1)
= 2(x^2 + 1) - 3 = 2x^2 - 1
Different results -> order matters.One-to-one: a function that can be undone
A function is one-to-one when different inputs always give different outputs — no output is ever shared. Only one-to-one functions can be reversed, because to run backwards from an output you need it to point to a single input. The graphical test is the horizontal line test: a one-to-one graph is met by every horizontal line at most once.
Building and checking an inverse
The inverse f⁻¹ undoes f: it swaps the roles of input and output, so the domain and range trade places. To find a formula, write y = f(x), swap x and y, then solve for y by isolating the variable.
f(x) = 2x - 3 1) y = 2x - 3 2) swap: x = 2y - 3 3) solve: x + 3 = 2y -> y = (x + 3)/2 So f^-1(x) = (x + 3)/2 Check that they undo each other: f(f^-1(x)) = 2 * (x+3)/2 - 3 = (x + 3) - 3 = x f^-1(f(x)) = ((2x - 3) + 3)/2 = (2x)/2 = x ✓