Functions

A rule that turns each input into exactly one output. Same input → same output.

Put the number in for x, in brackets, and do the arithmetic. f(3) with f(x)=3x+1 is 3(3)+1 = 10.

(-1)² = +1, not -1. Squaring kills the minus. Brackets every time you sub in.

A couple is the pair (x, f(x)) — the point you’d plot.

Domain = inputs. Range = outputs you actually get. Codomain = the bigger set the outputs sit inside.

Every input used, and each used only once. One arrow leaving each input. Outputs may repeat or go unused.

Set the rule equal to the given value and solve like any equation. Leave fractions exact.

Compute g(number) first — it’s just a number — then solve f(x) = that.

Set the two rules equal, gather x’s on one side, numbers on the other, solve.

Sub a known input–output pair into the rule and solve for k or a.

fg(x) = f(g(x)). The inner function (touching the x) runs first, the outer second.

Replace x in the outer with the whole inner expression, in brackets. Multiply out and tidy. Check by subbing a number.