Secondary 5 Step Functions Worksheet

All 7 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Finding the Rule of a Step Function (Greatest Integer Function)

   The graph of a step function is made of horizontal segments of length $2$ whose heights increase by $3$ from one segment to the next. The segment covering $4\le x<6$ sits at the height $y=5$, and every segment includes its left endpoint (closed dot on the left, open dot on the right).

   1. Find a rule of the form $f(x)=a\lfloor b(x-h)\rfloor +k$.
   2. Evaluate $f(11.4)$ and $f(3.5)$.
2. Q2Graphing a Step Function (Greatest Integer Function)

   Consider $f(x)=-2\lfloor x+1\rfloor +3$, restricted to $-2\le x<4$.

   1. Give the length and the height of each step, and say which endpoint of each step is closed.
   2. Sketch the graph on the grid below.
   3. Evaluate $f(2.5)$ and $f(-0.5)$.

   A blank Cartesian grid for this question is on the printable PDF.

3. Q3Solving Problems Involving the Step Function (the Greatest Integer Function)

   A ferry carries at most $24$ vehicles per crossing. The number of crossings needed to move $n$ vehicles ($n\ge 1$) is $$N(n)=\lfloor \frac{n-1}{24}\rfloor +1.$$

   1. How many crossings are needed for $100$ vehicles?
   2. For which numbers of vehicles are exactly $7$ crossings needed?
4. Q4The Inverse of the Step Function (Greatest Integer Function)

   Let $f(x)=3\lfloor x-1\rfloor +2$.

   1. Find every value of $x$ whose image is $5$.
   2. Deduce why the inverse of a step function is never a function.
   3. Describe the set of values on which the inverse relation of $f$ is defined.
5. Q5The Properties of the Step Function (Greatest Integer Function)

   Let $f(x)=-2\lfloor x+3\rfloor +1$.

   1. Evaluate $f(0)$, $f(-1.5)$ and $f(2.7)$.
   2. Give the length and the height of each step, and say whether $f$ is increasing or decreasing.
   3. Describe the range of $f$.
6. Q6The Role of the Parameters in a Step Function (Greatest Integer Function)

   Consider $f(x)=-3\lfloor \frac{1}{2}(x-1)\rfloor +4$. Without graphing, determine:

   1. the length of each step and the vertical distance between two consecutive steps;
   2. whether the steps go up or down as $x$ increases;
   3. whether the solid (included) endpoint of each step is at its left or its right end — justify, then confirm your answer by computing $f(1)$ and $f(3)$.
7. Q7The Step Function (Greatest Integer Function)

   Evaluate $\lfloor -3.2\rfloor$, $\lfloor 4\rfloor$ and $\lfloor -\frac{7}{2}\rfloor$. Then solve $\lfloor {\displaystyle \frac{x}{2}}\rfloor =3$ and write the solution set in interval notation.

The 5 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The definition, and the one place it surprises everybody

⌊x⌋ is the greatest integer that is less than or equal to x. Read it as "round down", never as "cut off the decimals" — for positive numbers the two agree, and for negative numbers they do not.

One consequence worth stating early: the brackets are not linear. You may not move a coefficient across them, and ⌊2x⌋ is not 2⌊x⌋. Whatever is inside stays inside, and is evaluated completely before the bracket acts.

What each parameter does to the staircase

The general rule is

and the four parameters split the description of the graph into four independent answers. This is the content of Q2(a), Q5(b) and Q6, and it is worth knowing cold.

h and k only slide the whole staircase sideways and vertically. They change where the steps start and how high they sit; they cannot change the length, the height, the direction or which dot is filled.

Evaluating: strictly inside out

Q1(b), Q2(c) and Q5(a) are pure evaluation, and the only way to get them wrong is to floor at the wrong moment. Compute the whole argument first, apply the bracket to that one number, then multiply by a and add k. For f(x) = −2⌊x + 3⌋ + 1 at x = 2.7:

Flooring 2.7 first and adding 3 afterwards would give a different, wrong answer. The bracket is applied to everything the notation encloses, and to nothing outside it.

Solving a floor equation: the answer is an interval

This is the technique behind Q3(b), Q4(a) and Q7, and it rests on one equivalence:

Closed on the left, open on the right, always. Isolate the bracket first — move k across, divide by a — then apply the equivalence, then solve the resulting double inequality for x. Every operation is performed on all three parts at once. For ⌊x ⁄ 3⌋ = 2:

The range is a set of numbers, not an interval (Q5)

Because ⌊…⌋ takes every integer value and nothing in between, a⌊…⌋ + k takes the values a·n + k as n runs through the integers — an evenly spaced, discrete set, with gap |a| between neighbours. That is the whole staircase seen edge-on.

So the correct way to write a range here is set-builder notation or an explicit listing with dots, never an interval. A function like −2⌊x + 3⌋ + 1 produces only odd integers, because doubling anything and adding 1 does. The domain, by contrast, is usually all of ℝ unless the question restricts it, as Q2 does — a step function accepts every real input and simply returns one of countably many outputs.

Building the rule from a described graph (Q1)

Q1 gives you the staircase in words and asks for the algebra. Work the four parameters in the same order you read them.

The step length gives |b| as its reciprocal, and the step height gives |a|. Which endpoint is closed then fixes the sign of b: left-closed means positive. Whether the heights increase or decrease fixes the sign of a relative to b. That leaves h and k, and the trick is to choose h to be the left edge of a step you were told about — then on that step the bracket evaluates to 0, and k is simply that step's height.

Finish by testing the neighbouring step: it should come out exactly one step-height away. That check catches a wrong h immediately, and it takes one line.

Sketching a staircase (Q2)

Find the boundaries by asking where the argument hits an integer, then draw one horizontal segment per step at the height you compute for it. Two conventions carry marks.

Step models in context (Q3)

Step functions describe anything charged or counted in whole blocks: crossings, fares, started hours, packets. The signature is a quantity that stays flat and then jumps, and the modelling question is always the same — where exactly does the jump fall?

Q3's rule counts how many full loads are needed. The −1 inside and the +1 outside are there so that an exact multiple of the capacity does not trigger an extra crossing: a load exactly filling the ferry should be the last crossing, not the first of a new one. That is a general pattern worth recognising, and it is why part (b) asks for the numbers of vehicles giving a fixed answer — the reply is a range of values, found with the interval equivalence above and then narrowed to whole numbers, since a count of vehicles is an integer.

Related idea for pricing by "each block begun": that convention needs steps closed on the right, because the last instant of a block still costs the old price and the very next instant starts a new one. By step 4 above, right-closed steps require a negative b — which is how a real-world phrase turns into the sign of a parameter.

Why the inverse is never a function (Q4)

Q4(a) asks which inputs map to a single given output, and the answer is an entire interval — that is the finding, not an inconvenience. Part (b) then wants the consequence spelled out: swapping the coordinates would assign to that one number every value in that interval, so it hands one input infinitely many images and fails the definition of a function. It is not a fixable defect either, because every step is a horizontal segment, so it happens at every output value.

Part (c) is about where the inverse relation is defined at all. A relation's inverse is defined exactly on the range of the original, and that range is the discrete set described above — so the inverse accepts only those isolated values and returns a half-open interval for each one. Restricting the domain, which rescues the inverse of an absolute value or a parabola, cannot help here: no restriction makes a horizontal segment one-to-one.

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Getting the most out of it

Do every negative bracket twice

Whenever a bracket's argument comes out negative, write the two integers on either side of it and pick the smaller one deliberately. Negative arguments are where nearly all lost marks on this topic live, and they turn up inside otherwise easy evaluation questions.

Write the double inequality before you touch x

On every floor equation, get to the line "n ≤ argument < n + 1" before doing any algebra on x. Once that line is on the page the rest is ordinary solving, and you cannot accidentally produce a single number as your answer.

Describe a graph in four sentences from memory

After Q6, cover the sheet and say the four facts about a staircase — length, height, direction, which dot is filled — and which parameter each comes from. Exams ask for those four in words far more often than they ask for a sketch.

Common questions