Functions and Graphs on NET tests five things: domain and range, one-to-one and onto, inverses, composition, and graph transformations. None of it needs long working. Most questions fall in under 40 seconds once you know the rules for each.
Key facts
- Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
- Functions and Graphs is unit 2 of the Class 11 PECTAA mathematics book (National Curriculum 2023)
- Our chapter map estimates about 6 MCQs from this chapter. NUST does not publish chapter counts
Source: NUST subject weightings, verified 22 August 2026; SailToNUST chapter map
What the chapter covers
The chapter defines a function as a rule that gives each input exactly one output. From there it asks four questions about any function. Which inputs are allowed (domain)? Which outputs come out (range)? Can you run it backwards (inverse)? And what happens when you feed one function into another (composition)?
The last part is about graphs: how shifting, stretching and reflecting a formula moves its graph. MCQs on it give a graph and ask which equation it belongs to, or the reverse.
Our practice set for this chapter also carries questions on sets, logic and binary operations. They are quick marks, and an hour of revision covers them.
Rules and formulas
Domain rules. Find what breaks the formula and exclude it:
sqrt(g(x)) needs g(x) >= 0
1 / g(x) needs g(x) ≠ 0
ln g(x) needs g(x) > 0
Inverse and composition:
To find f^-1: write y = f(x), swap x and y, solve for y
f(f^-1(x)) = x and f^-1(f(x)) = x
(f o g)(x) = f(g(x)) usually ≠ g(f(x))
For f(x) = (ax + b)/(cx + d):
f^-1(x) = (dx - b)/(a - cx)
Graph transformations, for y = f(x):
y = f(x) + k shift up by k
y = f(x - h) shift right by h
y = -f(x) reflect in the x-axis
y = f(-x) reflect in the y-axis
y = a f(x) stretch vertically by a
Even and odd: f(-x) = f(x) means even, with the graph symmetric about the y-axis. f(-x) = -f(x) means odd, with the graph symmetric about the origin.
Counting functions between finite sets, with n(A) = m and n(B) = n:
all functions A -> B = n^m
one-to-one functions A -> B = nPm (needs m <= n)
Worked MCQs
These five mirror the question types in our practice bank. We worked each one through and checked the result by substitution.
1. Find the domain of y = sqrt(-x).
The square root needs -x >= 0, so x <= 0. Domain: (-infinity, 0]. Students who see a minus sign and assume "no real domain" lose this one.
2. If f(x) = (3x + 1)/(2x - 1), find f^-1(x) and f^-1(2).
Use the shortcut with a = 3, b = 1, c = 2, d = -1:
f^-1(x) = (-x - 1)/(3 - 2x) = (x + 1)/(2x - 3)
f^-1(2) = 3/1 = 3
Check: f(3) = 10/5 = 2. Correct.
3. If h(x) = 3x + 2 and h(f(x)) = x, find f(2).
h(f(x)) = x means f is the inverse of h. So f(x) = (x - 2)/3 and f(2) = 0.
4. If f(x) = x^2 + 1 and g(x) = 2x - 3, find f(g(2)) and g(f(2)).
g(2) = 1, so f(g(2)) = f(1) = 2
f(2) = 5, so g(f(2)) = g(5) = 7
The two answers differ, and MCQs often list both as options. Work from the inside out.
5. Let L = {1, 2, 3}, M = {1, 2, 3, 4, 5, 6} and f(x) = 2x from L to M. What type of function is f?
The images are 2, 4 and 6. No two inputs share an output, so f is one-to-one. The values 1, 3 and 5 in M are never hit, so f is not onto. Answer: injective only.
Common traps
- Range from the formula alone. For
f(x) = x^2 + 2x + 5, complete the square:(x + 1)^2 + 4. The range is[4, infinity), not all real numbers. - Direction of a horizontal shift.
f(x - 2)moves the graph right, not left. Plug in the vertex to check:y = |x + 1| - 2has its vertex at (-1, -2). - Order of composition.
f(g(x))applies g first. Read it from the inside. - Onto depends on the codomain. The same rule can be onto one set and not onto a bigger one. Always check against the set the question names.
- Inverse exists only for one-to-one functions.
f(x) = x^2on all real numbers has no inverse. Restricted tox >= 0, it does.
How to prepare
This is a short chapter with a high return. Spend a session on the domain rules and the transformation table, then do a timed set. The ideas come back in limits and continuity, where a function that is undefined at a point is the whole question, and in the inverse trigonometric functions chapter.
If you are building your maths plan chapter by chapter, the NET mathematics guide puts this chapter in context with the rest of the syllabus.
FAQ
Are sets and binary operations included in NET maths?
Our practice bank for this chapter includes set, logic and group questions alongside the function questions. They take little time to learn, so cover them with the chapter rather than skip them.
How do I find the range of a function quickly?
For a quadratic, complete the square and read off the vertex. For a fraction like (ax + b)/(cx + d), every value except a/c is in the range, as long as c is not zero.
What is the difference between one-to-one and onto?
One-to-one means no two inputs give the same output. Onto means every element of the codomain is hit. A function that is both is a bijection and has an inverse.
