Functions
Functions
A function assigns exactly one output to each input. Master function notation, domain and range, composition, inverses, and transformations. Domain restrictions come from denominators (nonzero), even roots (radicand nonnegative), and logarithms (argument positive). For f(g(x)), the inner function's output must lie in the outer function's domain, so combine restrictions. Inverses undo each other: swap x and y, solve, and remember that the inverse's domain is the original's range. A function has an inverse function only if it is one-to-one; restrict the domain if needed, for example y = x^2 for x >= 0. Transformations follow a*f(b(x-h))+k: vertical stretch by |a|, horizontal compression by 1/|b|, shift right by h, up by k. Horizontal changes act opposite to the sign inside. For exponentials and logarithms, y = a^x and y = log_a(x) are inverses; the log graph has a vertical asymptote at x = 0. Average rate of change is (f(b)-f(a))/(b-a), not the slope of a curve at a point.
Traps: composing in the wrong order, forgetting to swap domain and range for inverses, applying a horizontal shift in the same direction as the sign, and losing domain restrictions after simplifying.
Worked example: Find the inverse of f(x) = 2^(x-3)+1. Swap: x = 2^(y-3)+1. Then x-1 = 2^(y-3), so y-3 = log_2(x-1), and f^-1(x) = log_2(x-1)+3, with domain x > 1.