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A quick, calm intro — then you choose when to climb.
A function is a machine with one rule: one input goes in, exactly one output comes out. means "take x, double it, add 3." The domain is the list of allowed inputs; the range is every output the machine can produce.
Every step explained — why we do it, and what happens.
Why: Read the rule out loud: "the machine takes x, doubles it, then adds 3." That is the whole function.
What happens: We know the recipe. Now feed it an input.
Why: f(4) means "run 4 through the machine." Replace every x with 4 — nothing else changes.
What happens: The input is now a real number, so the machine can start.
2 × 4 = 8 — multiplication first.
8 + 3 = 11 — the output.
Why: Label the answer so anyone can read it back: "the function of 4 is 11."
What happens: 4 → 11. One input, one output — the machine works.
A water tank starts with 100 litres, and a pump adds 40 litres every minute. Its volume follows the function f(t) = 40t + 100. How many minutes until the tank holds 340 litres?
Why: Translate the story into function form: t is the input (minutes), and the rule is "40 litres per minute, plus the 100 already in the tank."
What happens: t is the input; f(t) is the volume.
Why: We want the volume to be 340, so set the output equal to 340 and solve for the input.
What happens: Now it is a familiar balance equation.
Subtract 100 from both sides.
Divide both sides by 40: 240 ÷ 40 = 6 minutes.
Why: The check: run 6 back through the machine and see if it really outputs 340.
What happens: 40 × 6 + 100 = 340 ✓ — after 6 minutes the tank holds exactly 340 litres.
Why: The domain: can t be negative? "Minus 2 minutes" makes no sense for a real tank.
What happens: Only inputs t ≥ 0 are allowed — the domain protects the real-world meaning.
One classic glitch:
1.Feeding the machine an input it cannot handle
Wrong:Evaluating for — division by zero, the machine crashes
Right:Check the domain first. is not allowed, so it is not in the domain.
You've got the concept! Let's lock it in with practice.
Learning is optional — the ladder is always open.