Evaluating a Composite Function at a Value (F 604)

Think of f and g as two machines wired in series. A number goes into the first machine, g, and whatever pops out gets fed straight into the second machine, f. You cannot run the second machine until the first one has finished, so the order of operations is forced: inside first, outside second. That inside function is literally sitting inside the parentheses of the outside function, which is your visual cue for which one runs first.

Read f(g(3)) by starting with the innermost piece, g(3). Compute that number completely. Then take whatever value you got and treat it as the new input to f, writing f(that number). You are never evaluating f and g independently and combining them at the end — the output of g becomes the actual x-value you substitute into f's rule. This is different from a left-to-right reading, which would tempt you to deal with f first simply because its letter appears first in the notation.

This idea sits right next to transformations and inverses because all three are about tracking what a function does to an input. A transformation changes the rule for one machine; an inverse asks what input would produce a given output by running the machine backward; composition chains two separate machines together. Once evaluating f(g(x)) at a specific number feels automatic, the same inside-out logic carries over to composing two transformed graphs or undoing a composition step by step.