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Lesson 16 min· Section 1 of 4

The straight-up lesson, built for the exam.

Functions

Formula sheet

Everything for this chapter, in one place.

Composition


Meaning	$(f \circ g)(x) = f(g(x))$ — inner function first
Order	$f(g(x)) \neq g(f(x))$ in general
Domain of $f(g(x))$	$x$ must be allowed in $g$ , and $g(x)$ must be allowed in $f$

Functional equations and what they force


$f(x+y) = f(x)f(y)$	$f(x) = a^x$ , so $f(n) = f(1)^n$
$f(xy) = f(x)f(y)$	multiplicative; always $f(1) = 1$ (or 0)
$f(x+y) = f(x) + f(y)$	$f(x) = cx$
$f(x) + k\,f\!\left(\frac{1}{x}\right) = g(x)$	write it again with $x \to \frac{1}{x}$ and solve the two together
$f(x+2) = f(x) + f(x+1)$	a recurrence — step forward from two known values
$f(xy) = f(x)f(y) + f(x) + f(y)$	set $g = f+1$ ; then $g$ is multiplicative

Useful substitutions


$x = y = 1$	pins down $f(1)$
$y = x$	turns a two-variable law into one variable
$x \to \frac{1}{x}$	pairs with the original to eliminate the unwanted term
A prime factorisation	a multiplicative $f$ is determined by its values on primes

Symmetry


$f(a+x) = f(a-x)$ for all $x$	the graph is symmetric about $x=a$
So the roots pair up	each pair sums to $2a$
With $2k$ roots	their total is $2ak$

The larger or smaller of two curves


$\min\{u(x), v(x)\}$	its maximum sits where the two curves cross
$\max\{u(x), v(x)\}$	its minimum sits where they cross
Either way	solve $u(x) = v(x)$ and substitute

Range of a rational function


Method	set $y = \dfrac{\text{num}}{\text{den}}$ , cross-multiply into a quadratic in $x$
Then	real $x$ needs discriminant $\ge 0$ — that inequality is the range
Alternative	substitute for a repeated block and track what it can be

Greatest integer, $[x]$


Meaning	the largest integer not exceeding $x$
So	$[x] = n \iff n \le x < n+1$
Summing $[\,\cdot\,]$ over a range	group the terms that share a value, then count each group
$\lfloor\sqrt{n}\rfloor = k$	for the $2k+1$ values $n = k^2$ to $k^2+2k$

Why this chapter matters

0.8

questions in a typical Quant section

15th

of 25 chapters

≈ 2 of 66 marks a section ride on it.

Asked in20 of 24 papers

20172025

Year by yearper section

2017: 12017: 1 per section
2018: 12018: 1 per section
2019: 12019: 1 per section
2020: 0.62020: 0.6 per section
2021: 0.72021: 0.7 per section
2022: 12022: 1 per section
2023: 0.32023: 0.3 per section
2024: 12024: 1 per section
2025: 12025: 1 per section

Question type23 asked

MCQ 52%Type-in 48%

Type-in answers carry no negative marking.

From every Functions question in CAT papers since 2017.