The range is not the y-axis

tracing the curve

n/ax, the input

n/af(x), the output

in the domainverdict

The shading on the x-axis is the domain. The shading on the y-axis is the range. They are different axes and different questions.

The figure needs JavaScript. It plots f(x) = √(2 − x), shading the domain x ≤ 2 on the x-axis and the range f(x) ≥ 0 on the y-axis, with f(2) = 0, f(1) = 1, f(−2) = 2 and f(−7) = 3. A later view reflects the curve in y = x to give the inverse 2 − x², where (−7, 3) becomes (3, −7). Everything below stands on its own.

Two reasons a domain stops

Reason	Example	What the domain is
The arithmetic forbids it	f(x) = √(2 − x)	x ≤ 2, because a square root needs 2 − x ≥ 0
The situation forbids it	C(n) = 8000 + 150n, a coach	n a whole number from 1 to 45, because that is a coach

The second kind is the one modelling questions are about, and the one that gets left out. Nothing in 8000 + 150n stops you putting in n = 2.5 or n = 1000. A coach does.

The square root: where the arithmetic stops you

f(x) = √(2 − x).

Domain. Need 2 − x ≥ 0, so x ≤ 2. Note the direction: subtracting x flips the inequality, and that is where it goes wrong.
Range. A square root is never negative, and it can be made as large as you like by going far enough left. So f(x) ≥ 0.
Check the ends: f(2) = 0, f(1) = 1, f(−2) = 2, f(−7) = 3, f(−14) = 4.

Notice how slowly the outputs climb. To get f(x) up to 4 you have to go all the way to x = −14. That shape is the square root's signature and it is worth recognising without plotting.

"All real numbers" is almost never a range. It is the answer students give when they have not looked. Ask instead: what is the smallest output, what is the largest, and can the function reach them? Here the smallest is 0 and it is reached, at x = 2. There is no largest.

The coach: where the situation stops you

A coach for a school trip costs 8,000 baht plus 150 baht a student, so C(n) = 8000 + 150n, and the coach holds 45.

Domain: n ∈ {1, 2, 3, …, 45}. Whole numbers, because students are people, and at most 45, because that is the coach.
Range: the 45 values C(1) = 8150, C(2) = 8300, … up to C(45) = 14750, each 150 apart.

The range is not "8150 to 14750". No trip costs 9000 baht: C(n) = 9000 would need n = 6.67. Writing an interval describes a box the answers sit in, which is a different claim and a weaker one.

Function notation is doing work here. C(n) says the cost depends on the number of students and nothing else. When a question asks for C(20) it is asking one specific thing: 11000. When it asks you to solve C(n) = 12050 it is asking the reverse, and the answer is n = 27. Same function, opposite direction, which is what an inverse formalises.

The inverse, informally

An inverse undoes the function. For f(x) = √(2 − x):

Write y = √(2 − x), swap the letters, solve for y: f⁻¹(x) = 2 − x²
Its domain is the original's range, so x ≥ 0. Its range is the original's domain, so f⁻¹(x) ≤ 2.
Check: f(−7) = 3, and f⁻¹(3) = 2 − 9 = −7. Back where we started.

On a graph the inverse is the reflection in the line y = x, and that is why the coordinates swap: (−7, 3) becomes (3, −7). A point already on y = x does not move, which is why (1, 1) sits on both curves.

Only one-to-one functions have inverses. f(x) = x² does not, on its own: f(3) and f(−3) are both 9, so f⁻¹(9) would have to be two things at once. Restrict the domain to x ≥ 0 and it does. At Standard Level you need to know that, and restricting the domain yourself is Higher Level, in 2.7.

On the GDC: reading a range off a graph

A range is a question about the picture, so the fastest route is to plot it and look. The machine will not tell you the range; it will show you enough to say it.

When you may use it. Applications allows a calculator in every paper. Analysis does not allow one in Paper 1, so you must be able to reason a domain and range out from the formula as well.

TI-Nspire CX II

ctrl doc → Add Graphs, then ctrl x² for the root and 2-x inside it
menu → Window / Zoom → Window Settings, and set x from -15 to 5
The curve stops dead at x = 2. That full stop is the domain, drawn.
menu → Trace → Graph Trace and walk along it, watching the y readout never go below 0

Casio fx-CG50

MENU → Graph, then SHIFT x² for the root and 2-X
SHIFT F3 V-Window, Xmin -15, Xmax 5, Ymin -1, Ymax 5
F6 DRAW, then SHIFT F1 Trace and walk along it
Try to trace past x = 2 and nothing is there, which is the domain telling you where it ends

The mark people lose. Giving the range as an interval of the window you happened to choose. Set Ymax to 5 and the curve fills the screen, so a student writes "0 to 5". Go to x = −100 and f(x) is just over 10. The range has no upper end, and no window will show you that; only the formula will. The other one is writing the domain and the range the same way round, so check which axis you are describing before you write a single inequality.

Your turn

1. For f(x) = √(2 − x), what is f(−14)?

2. For the coach, C(n) = 8000 + 150n, what is C(20), in baht?

3. Solve C(n) = 12050. How many students?

4. For f(x) = √(2 − x), what is f⁻¹(3)?

5. A student gives the range of the coach cost as "8150 ≤ C ≤ 14750". What is wrong with it?

Question 5. A student gives the range of the coach cost as 8150 to 14750. What is wrong with it? Nothing; those are the smallest and largest costs. The range is 45 separate values 150 apart, not every number between them. The upper limit should be 14900. It has given the domain instead of the range.

Where the marks go

1 markThe domain, with the right inequality direction, or the right set.

1 markThe range, as a statement about outputs.

1 markFor an inverse, the correct expression.

1 markAnd its domain, which is the original's range.

In a modelling question the domain mark is usually for the context restriction, not the algebraic one, and it is the mark most often left blank. If a question is about people, buses or days, say so.