Common content, so this runs for both courses.
Ask for the range of √(2 − x) before the domain. Doing it the usual way round, domain first, trains the habit of answering a question about x and then answering it again. Ask for the range cold and you will get "all real numbers" from most of the room, which is the answer to neither question.
Then step the figure. The green bar grows along the x-axis and the orange bar grows up the y-axis, and the point is that they are two different axes and two different questions. Stop at x = −14 and say the useful thing: the output is only 4, there is no largest output, and no calculator window will ever show you that.
The last view is the reflection. Ask which point does not move before you show them, and someone will spot (1, 1).
| Question | Answer |
|---|---|
| 1. f(−14) | √(2 + 14) = √16 = 4. |
| 2. C(20) | 8000 + 3000 = 11000 baht. |
| 3. C(n) = 12050 | 150n = 4050, so n = 27. |
| 4. f⁻¹(3) | 2 − 9 = −7. |
| 5. The interval range | B. 45 separate values, 150 apart. |
1 markThe domain, with the inequality the right way round, or the right set.
1 markThe range, as a statement about outputs.
1 markFor an inverse, the expression.
1 markAnd its domain, which is the original's range.
In a modelling question the domain mark is for the context restriction, not the algebraic one. A student who writes "x ∈ ℝ" for the number of students on a coach has answered a question about algebra in a question about a bus.
| They wrote | What happened |
|---|---|
| x ≥ 2 | Solved 2 − x ≥ 0 without flipping the inequality. The single commonest error in this sub-topic. Ask them to test x = 3. |
| "All real numbers" for the range | Did not look. Ask for the smallest output and whether it is reached. |
| 16 on question 1 | Stopped at what goes under the root. |
| 3.46 | Used 2 − 14 instead of 2 − (−14). A sign inside a bracket. |
| 3000 on question 2 | Left out the fixed cost. They have read the rate and not the model. |
| 550 | Gave the cost per student. A good number and the wrong question. |
| 163000 | Multiplied the whole of C(1) by 20. The fixed cost is paid once. |
| 80.33 | On question 3, divided 12050 by 150 without removing the 8000. |
| 7 on question 4 | 2 − 9 read as 7. Worth noting that f(7) does not exist either, so the answer is impossible twice over. |
| 1.73 | On question 4, applied f rather than its inverse, and to a value outside the domain. |
"Why does the inequality flip?" It does not flip because of a rule about square roots; it flips because you divided or multiplied by −1. Do it in two visible steps: 2 − x ≥ 0, then 2 ≥ x. No flip needed if you move x to the other side instead, which is the version worth teaching.
"Can the range have gaps?" Yes, and the coach is the example: it has 44 of them. This is also where a piecewise model comes from later, in AHL 2.9.
"Does f⁻¹ mean one over f?" No, and it is the worst notation in the course. Point out that f⁻¹(3) = −7 here, while 1/f(3) does not exist at all because 3 is outside the domain. The two are not just different, one of them is undefined.
| Stage | What to do |
|---|---|
| Demonstrate | Plot it, then try to trace past x = 2. Nothing is there. That full stop is the domain, drawn, and it is more convincing than an inequality. Then set Ymax to 5 and ask for the range; somebody will say "0 to 5". Change Ymax to 12 and ask again. |
| Where they stick | Entering √(2−x) without brackets, so the machine draws √2 − x, a straight line with a gap at nothing. If the graph looks linear, the brackets are missing. |
| The check | Before plotting, ask what f(3) should be. "Error" is the right answer, and a student who expects it has understood the domain. One who expects a number will not notice when the machine disagrees with them. |
Set equal scales on the window before showing the inverse, or the reflection in y = x looks like a reflection in something else and the one visual idea in the sub-topic is lost.
| Step | What |
|---|---|
| 1 | "What is the range of √(2 − x)?" Cold. Collect "all real numbers". |
| 2 | Step the figure. Two bars, two axes, two questions. |
| 3 | The domain from the arithmetic, in two steps so nothing flips. |
| 4 | The coach. Domain from the situation, and a range with gaps in it. |
| 5 | Function notation forwards, C(20), then backwards, C(n) = 12050. |
| 6 | The inverse as the formal version of backwards, and the reflection. |
| 7 | One-to-one, with x² as the example that fails. Flag 2.7 for HL. |
Do not say "the domain is the x values and the range is the y values". It is true and it is the reason students answer the same question twice. Say "the domain is what you are allowed to put in, the range is what can come out", which keeps the direction in the sentence.
Do not let a window stand in for a range. If you read a range off a screen once, every student will do it for the rest of the course, and on an unbounded function they will be wrong every time.