Function Notation & Evaluation

Calculus Readiness · Axiom Academy

Reading and using f(x) — a rule named once, fed an input, returning an output. It is a machine, not a multiplication. Picture a function as a machine: you drop an input in the top, the rule inside processes it, and an output comes out the bottom. The rule below is "double it, then add 3." Feed it 5 and you get 13 . f(5)=13 is the output that drops out 2. Evaluating = Substituting Everywhere To evaluate f(a) , replace every x in the rule with a , then simplify. Watch the rule 3x^2 - 5x + 1 as each x is overwritten by (2) — both copies at once — and the expression collapses to a single number. With f(x) = 3x^2 - 5x + 1 , find f(2) : f(2) = 3(2)^2 - 5(2) + 1 = 3(4) - 10 + 1 = 3 . So f(2) = 3 . The two copies of x became two copies of (2) — miss one and the answer is wrong. The input need not be a number. With f(x)=x^2+4 , f(x+1) = (x+1)^2 + 4 = x^2 + 2x + 5 . f(x+1) is not f(x)+1 . The first feeds a new input in; the second adds 1 to the old output. Here f(x)+1 = x^2+5 — a different result. For f(x)=x^2 , f(-3) = (-3)^2 = 9 , not -3^2 = -9 . The parentheses carry the sign into the square. For f(x)=x^2-4 : evaluating f(5)=21 gives one output; solving f(x)=5 gives the inputs . A graph is the same machine drawn as a picture: pick an input a on the horizontal axis, go straight up to the curve, then straight across to the vertical axis. Where you land is f(a) . Below, the input a=2 on reads off the output f(2)=2 .

This is the written version of the interactive lesson above. See the full Calculus Readiness course.