Functions — NDA Mathematics Formulas and Traps

What a Function Is, and How to Classify It

Counting functions of a given type

Number of functions A → B

\text{functions }A\to B=n^{m}\qquad \text{injections}={}^{n}P_{m}=\dfrac{n!}{(n-m)!}\qquad \text{bijections }(m=n)=n!\qquad \text{relations}=2^{mn}

Common traps

Piecewise rules must agree at the boundary

A two-piece rule like

f(x)=x^2

on

[0,4]

and

3x

on

[4,8]

is only a function if the pieces give the same value at the shared point

x=4

(here

16\neq12

, so it is not well-defined). Always check the join before declaring it a function.

'Onto' is not absolute — it depends on the codomain

f:\mathbb{N}\to\mathbb{N}

,

f(x)=x+1

is one-one but not onto (nothing maps to 1). The same rule on

\mathbb{Z}\to\mathbb{Z}

is onto. Read the declared domain and codomain before deciding.

A non-monotone function usually fails one-one (and often onto)

On a symmetric interval, an even-ish function doubles back:

\cos(\pi x)

on

(-1,1)

hits its maximum at

x=0

, so it is neither one-one nor onto

(-1,1)

. Contrast

x|x|

, which is strictly increasing and is a bijection of

(-1,1)

. Check monotonicity before declaring a bijection.

Domain, Range, and the Standard Properties

Learn this subtopic in the notes

Even and odd functions

Even and odd tests

f\text{ even}\iff f(-x)=f(x)\qquad f\text{ odd}\iff f(-x)=-f(x)

Periodic functions and their period

Period after scaling the argument

\text{period of }\sin(kx)=\dfrac{2\pi}{|k|}\qquad \text{period of }\tan(kx)=\dfrac{\pi}{|k|}\qquad \text{period of }f(ax+b)=\dfrac{T}{|a|}

Common traps

≥ 0 under a plain root, but > 0 when the root is a denominator

A root by itself allows equality (

\sqrt{g}

needs

g\ge0

). The moment that root is in a denominator — e.g.

\dfrac{1}{\sqrt{|x|-x}}

— the value 0 is banned too, so you need the strict inequality

g>0

. Missing this flips a closed bracket to an open one and loses the mark.

Range is not the codomain

If a question declares

f:\mathbb{R}\to\mathbb{R}

but the outputs only fill

[0,1)

, the range is

[0,1)

— not

\mathbb{R}

. 'Onto' questions are really 'shrink the codomain to the range' questions.

$f(0)=0$ is necessary for odd, not sufficient

Many odd functions pass through the origin, but passing through the origin does not make a function odd — you must verify

f(-x)=-f(x)

for all

x

. And a sum like even + odd is usually neither.

$|x|$ is even, never odd

|-x|=|x|

, so the modulus is even. Combinations like

|x|-x^3

mix an even and an odd part and end up neither. Also:

\sqrt{x^2}=|x|

, not

x

— a common sign slip.

Composition and Inverse of Functions

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Composition of functions

Composition and its inverse

(f\circ g)(x)=f(g(x))\qquad (f\circ g)^{-1}=g^{-1}\circ f^{-1}

When do two linear functions commute?

Commuting condition for linear f, g

f\circ g=g\circ f\iff b(c-1)=d(a-1)\ \ [f=ax+b,\ g=cx+d]

Inverse of a function

Inverse of a linear function

f(x)=ax+b\ (a\neq0)\ \Rightarrow\ f^{-1}(x)=\dfrac{x-b}{a}

Common traps

$(fg)$ is a product, $(f\circ g)$ is a composition

NDA writes the product as

(fg)(x)=f(x)g(x)

and the composition as

(f\circ g)(x)=f(g(x))

. They are completely different operations — read the symbol carefully before computing.

Work inside-out, and keep the order

(f\circ g)(x)=f(g(x))

means

g

acts first. Students often apply

f

first or read

f\circ g

as a product. For iterated composition, peel one layer at a time — don't try to do all of

f\circ f\circ f

in one leap.

Inverse needs a bijection — and $f^{-1}\neq1/f$

Only one-one onto functions have an inverse;

x^2

on

\mathbb{R}

has none (not one-one). And

f^{-1}

is the undo function, not the reciprocal —

f^{-1}(x)

is generally nothing like

\dfrac{1}{f(x)}

. When finding it, remember the domain of

f^{-1}

is the range of

f

.

The Greatest Integer (Floor) Function

Learn this subtopic in the notes

Common traps

Floor rounds DOWN, so negatives go further from zero

[-1.3]=-2

, not

-1

: you must go to the integer below. Truncating toward zero is the most common floor mistake on negative inputs.

[x] = n is an interval, not a single point

Solving a floor-equation gives integer values of $[x]$ ; each one unpacks to a whole interval

[n,n+1)

of real

x

. Reporting only the integers

x=n

misses the rest of the solution set.

Functional Equations

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Multiplicative and additive forms

Signature functional-equation solutions

f(xy)=f(x)f(y)\Rightarrow f(x)=x^{k}\qquad f(x+y)=f(x)f(y)\Rightarrow f(x)=a^{x}\qquad f(x+y)=f(x)+f(y)\Rightarrow f(x)=cx

Common traps

One equation, two unknowns — make a second

You cannot read off

f(x)

from a single relation that also contains

f(1/x)

or

f(1-x)

. Generate the partner equation by substituting, then solve the

2\times2

system. Substituting a value that is its own partner (like

x=\tfrac12

for

x\to1-x

) can shortcut a single requested value.

Solve for the original variable before substituting

From

f(x+1)=x^2-3x+2

you must write

x=t-1

(where

t=x+1

) and plug that in — not simply replace

x

by

x

in the right side. Getting the shift direction backwards is the usual slip.

Functions formulas

What a Function Is, and How to Classify It

Counting functions of a given type

Common traps

Domain, Range, and the Standard Properties

Even and odd functions

Periodic functions and their period

Common traps

Composition and Inverse of Functions

Composition of functions

When do two linear functions commute?

Inverse of a function

Common traps

The Greatest Integer (Floor) Function

Common traps

Functional Equations

Multiplicative and additive forms

Common traps

More NDA Mathematics formula sheets