Relations and Functions: Notation, Transformations, and Inverses
Functions, Transformations, and Inverses
A function pairs each element of its domain with exactly one element of its range. Before graphing or computing, extract domain restrictions from the structure: denominators cannot be zero, even-indexed radicals need non-negative radicands, and logarithms need positive arguments. Function notation is an instruction, not decoration - f(2a) means substitute 2a wherever x appears, and f(g(x)) means compose, where g's outputs must lie inside f's domain or the composition is undefined there.
For y = a·f(b(x − h)) + k, the parameters outside the function act on y as written: vertical stretch by |a|, reflection in the x-axis when a < 0, and vertical shift k. Parameters inside act on x and behave in reverse: horizontal stretch by 1/|b|, reflection in the y-axis when b < 0, and shift right by h. Factor b out of the bracket before reading h.
To find an inverse, swap x and y and solve; the domain and range trade places. The vertical line test decides whether a relation is a function, and the horizontal line test decides whether its inverse will be - if it fails, restrict the domain (for example, y = x² taken on x ≥ 0).
Traps: sign errors on h; treating a horizontal stretch as vertical; assuming the domain of a composition equals the domain of the inner function; confusing f⁻¹(x) with 1/f(x).
Worked example: f(x) = √(x + 3) − 1 has domain x ≥ −3 and range y ≥ −1. Swapping gives x = √(y + 3) − 1, so y = (x + 1)² − 3. Thus f⁻¹(x) = (x + 1)² − 3, valid only for x ≥ −1.
Rule of thumb: outside changes follow y directly; inside changes follow x in the opposite direction.