Mathematics from zero
Functions: multiple-choice review
Crux Multiple-choice synthesis across the functions unit: the one-output rule, function notation, evaluation, linear rate and starting value, and reading a graph.
Your altitude — climbing toward senior
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You are at middle altitude — in the skySix questions that cut across the whole unit. Each one tests whether you can use a function, not just recite its definition — feed inputs in, read outputs out, and recognise the rule behind the picture.
Goal
Confirm you can connect the four ideas the lessons built: a function is a one-output-per-input rule, notation names that rule, a linear function is a steady rate plus a starting value, and its graph is a straight line.
Quiz
Completed
A machine returns 5 the first time you feed in 2, then returns 9 the next time you feed in 2. Is it a function, and why?
Heads-up Producing valid numbers is not enough. The defining rule is that one input maps to exactly one output, every time — two different outputs for input 2 disqualifies it.
Heads-up Same input is exactly the case the rule constrains: the same input must yield the same output. Here it did not, so it is not a function.
Heads-up You do not need a formula. A function can be a table or a machine; the test is behavioural — same input, same output. That already failed here.
Quiz
Completed
A function follows the rule f(n) = 3n + 4. What is f(5)?
Heads-up The 3 is attached to n, so it multiplies the input: 3 × 5 = 15, then + 4. Adding 3 and 5 first treats the rate as if it stood alone.
Heads-up The input is the whole point of evaluating a function. f(5) means feed 5 into the rule: 3 × 5 + 4. You cannot drop the input.
Heads-up Order matters: the number attached to n (the rate) multiplies the input, then the lone number is added. That is 3 × 5 + 4 = 19, not 3 × (5 + 4).
Quiz
Completed
A vending machine accepts only buttons A1, A2, and A3, and each drops one specific snack. What are its inputs (domain) and outputs (range)?
Heads-up A one-to-one mapping does not make the two sets identical. Buttons go in, snacks come out — different things. The domain is the allowed inputs; the range is the produced outputs.
Heads-up It is reversed. You press a button (input) and a snack comes out (output). The thing you supply is the input; the thing produced is the output.
Heads-up The button you press is the input — it is exactly what determines the output. A function always has inputs; here they are the accepted buttons.
Quiz
Completed
For the linear function y = 4x + 7, the input changes from 2 to 3. By how much does the output change, and why?
Heads-up The starting value is the output at input 0, not the per-step change. The per-step change is the slope — the number multiplied by the input, which is 4.
Heads-up Adding the rate and the starting value mixes two different roles. The step per input is just the slope, 4; the 7 sits there unchanged as the input moves.
Heads-up That is the whole point of linear: the step is constant. You already know it equals the slope, 4, without computing y at 2 and 3.
Quiz
Completed
On a function's graph there is a point at (6, 2). What does this point tell you?
Heads-up The pair order is fixed: first number is the input (across), second is the output (up). (6, 2) is input 6, output 2 — not the reverse.
Heads-up A point is a pair, not a multiplication. It records one input-output result the function produced, here 6 in and 2 out.
Heads-up A single point names one input and one output, not two outputs. The first coordinate is the input on the horizontal axis, the second is the output on the vertical axis.
Quiz
Completed
You plot the outputs of f(n) = 2n + 1 for n = 0, 1, 2, 3 and they land at (0,1), (1,3), (2,5), (3,7). Without plotting more points, what shape will the full graph be, and why?
Heads-up Getting bigger is not what bends a graph; a changing step is. Here the step is a fixed 2 each time, so the points stay perfectly straight.
Heads-up A linear rule applies the same constant step everywhere, not just between sampled points. The whole graph is one straight line.
Heads-up The constant step holds for every input, not just whole ones. The slope of 2 forces a straight line across the entire graph.
Recap
The through-line of the unit is one machine: an input goes in and exactly one output comes out, every time. Function notation like f(5) just names the rule and the input you feed it; the domain is what you may feed in, the range is what comes out. A linear function adds structure — a steady rate (the slope, the per-step change) plus a starting value (the output at input 0) — and because that step never changes, its graph is always a straight line, with each point read as (input, output).