NDA Maths · Teaching notes

Functions — NDA Mathematics

Functions is a reliable scoring chapter in NDA Mathematics — around 109 past-year questions across 2017–2026, roughly five or six marks on a typical paper, and only about one in ten is HARD. Most of it is bread-and-butter: read off a domain, find a range, test even/odd or periodicity, compose two functions, invert one. The marks are lost not to difficulty but to a handful of standard traps — forgetting a denominator restriction, assuming f∘g = g∘f, or mishandling the floor function near integers. Work the five notes below in order — first what a function is and how to classify it, then domain/range and the standard properties, then composition and inverse, then the greatest-integer function, and finally functional equations — and the bank turns into careful rule-application.

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Subtopic notes

PYQ weightage by concept

21 concepts · 109 PYQs — where the marks actually sit, so you know what to drill first

What a Function Is, and How to Classify It8 PYQs · 7%

Concept	PYQs	Share
One-one, onto, and bijective	5	5%
Is it a function? Well-defined and the vertical-line test	2	2%
Counting functions of a given type	1	1%
Domain, codomain, range — the vocabularyfoundation	—	—

Domain, Range, and the Standard Properties52 PYQs · 48%

Concept	PYQs	Share
Finding the range	17	16%
Finding the domain (roots, denominators, logs)	11	10%
Periodic functions and their period	7	6%
Standard functions and their graphs	7	6%
Even and odd functions	6	6%
The modulus function and distance	4	4%
Reading domain and range from a graphfoundation	—	—

Composition and Inverse of Functions30 PYQs · 28%

Concept	PYQs	Share
Composition of functions	16	15%
Evaluating and combining functions	6	6%
When do two linear functions commute?	4	4%
Inverse of a function	4	4%

The Greatest Integer (Floor) Function7 PYQs · 6%

Concept	PYQs	Share
Floor equations and sums	3	3%
Definition, graph, and behaviour at integers	2	2%
The fractional part {x} = x − [x]	2	2%

Functional Equations18 PYQs · 17%

Concept	PYQs	Share
Multiplicative and additive forms	11	10%
Solving by substitution (x → 1/x, x → 1−x)	5	5%
Undoing an argument shift	2	2%

Formula & revision sheet

3 formulas · 13 gotchas across all subtopics — the exam-eve cheat-sheet

What a Function Is, and How to Classify It

Formulas (1)

Counting functions of a given type · Number of functions A → B $|A|=m,\ |B|=n\ \Rightarrow\ \text{functions}=n^{m},\quad \text{injections}={}^{n}P_{m}$

Watch out for (2)

Piecewise rules must agree at the boundary
→ Is it a function? Well-defined and the vertical-line test
'Onto' is not absolute — it depends on the codomain
→ One-one, onto, and bijective

Domain, Range, and the Standard Properties

Formulas (1)

Periodic functions and their period · Period after scaling the argument $\text{period of }\sin(kx)=\frac{2\pi}{|k|},\qquad \text{period of }\tan(kx)=\frac{\pi}{|k|}$

Watch out for (4)

≥ 0 under a plain root, but > 0 when the root is a denominator
→ Finding the domain (roots, denominators, logs)
Range is not the codomain
→ Finding the range
$f(0)=0$ is necessary for odd, not sufficient
→ Even and odd functions
$|x|$ is even, never odd
→ The modulus function and distance

Composition and Inverse of Functions

Formulas (1)

When do two linear functions commute? · Commuting condition for linear f, g $f(x)=ax+b,\ g(x)=cx+d:\quad f\circ g=g\circ f \iff b(c-1)=d(a-1)$

Watch out for (3)

$(fg)$ is a product, $(f\circ g)$ is a composition
→ Evaluating and combining functions
Work inside-out, and keep the order
→ Composition of functions
Inverse needs a bijection — and $f^{-1}\neq1/f$
→ Inverse of a function

The Greatest Integer (Floor) Function

Watch out for (2)

Floor rounds DOWN, so negatives go further from zero
→ Definition, graph, and behaviour at integers
[x] = n is an interval, not a single point
→ Floor equations and sums

Functional Equations

Watch out for (2)

One equation, two unknowns — make a second
→ Solving by substitution (x → 1/x, x → 1−x)
Solve for the original variable before substituting
→ Undoing an argument shift