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Function formulas

Master Function through 36 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Function, every formula

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Number of functions A→B
$n (B)^{n (A)}$
One-one (injection)
$a_{1} \neq = a_{2} \Rightarrow f (a_{1}) \neq = f (a_{2})$
Q1MCQOne-one
A function is one-one if:
- A
  distinct inputs give distinct outputs
- B
  every element of $B$ has a pre-image
- C
  $f(A)=B$
- D
  it is periodic
Number of injections A→B
$^{n} P_{r}, r = n (A), n = n (B), r \leq n$
Onto (surjection) condition
$f (A) = B (range = codomain)$
Q1MCQOnto
A function $f:A\to B$ is onto iff:
- A
  range $=$ codomain
- B
  range $\subset$ codomain
- C
  it is one-one
- D
  $n(A)<n(B)$
Number of onto functions (n(B)=2)
$2^{n} - 2, n = n (A)$
Bijection: n(A)=n(B)
$f bijection \Rightarrow n (A) = n (B)$
Q1MCQBijection
If $f:A\to B$ is a bijection between finite sets, then:
- A
  $n(A)=n(B)$
- B
  $n(A)>n(B)$
- C
  $n(A)<n(B)$
- D
  $n(A)=2n(B)$
Number of bijections
$n! when n (A) = n (B) = n$
Composite function
$(g \circ f) (x) = g (f (x))$
Q1MCQComposite
If $f(x)=x+1$ and $g(x)=x^{2}$, then $(g\circ f)(2)$ is:
- A
  $9$
- B
  $5$
- C
  $4$
- D
  $8$
Composite not commutative
$f \circ g \neq = g \circ f (in general)$
Q1MCQNon-commutative
In general, $f\circ g$ and $g\circ f$ are:
- A
  not equal
- B
  always equal
- C
  both identity
- D
  inverses
Inverse of a composite
$(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}$
Q1MCQInverse of composite
$(g\circ f)^{-1}$ equals:
- A
  $f^{-1}\circ g^{-1}$
- B
  $g^{-1}\circ f^{-1}$
- C
  $f\circ g$
- D
  $g\circ f$
Inverse function condition
$f invertible \Leftrightarrow f is a bijection$
Q1MCQInvertibility
A function is invertible iff it is:
- A
  a bijection
- B
  only one-one
- C
  only onto
- D
  even
Inverse pair identities
$f \circ f^{- 1} = I_{B}, f^{- 1} \circ f = I_{A}$
Q1MCQInverse identities
$f\circ f^{-1}$ equals:
- A
  $I_B$ (identity on $B$)
- B
  $f^{2}$
- C
  $0$
- D
  $f$
Graph of f and f⁻¹
$y = f (x) and y = f^{- 1} (x) are symmetric about y = x$
Q1MCQGraph symmetry
The graphs of $y=f(x)$ and $y=f^{-1}(x)$ are symmetric about:
- A
  $y=x$
- B
  the x-axis
- C
  the y-axis
- D
  the origin
Even function
$f (- x) = f (x)$
symmetric about y-axis
Q1MCQEven function
Which function is even?
- A
  $f(x)=x^{2}$
- B
  $f(x)=x^{3}$
- C
  $f(x)=\sin x$
- D
  $f(x)=x$
Odd function
$f (- x) = - f (x)$
symmetric about origin
Q1MCQOdd function
Which function is odd?
- A
  $f(x)=x^{3}$
- B
  $f(x)=x^{2}$
- C
  $f(x)=\cos x$
- D
  $f(x)=|x|$
Even + odd decomposition
$f (x) = even \frac{1}{2} [f (x) + f (- x)] + odd \frac{1}{2} [f (x) - f (- x)]$
Q1MCQEven+odd split
Every function can be written as the sum of:
- A
  an even and an odd function
- B
  two even functions
- C
  two odd functions
- D
  two periodic functions
Modulus function
$∣ x ∣ = {x - x x \geq 0 x < 0$
domain R, range [0,∞)
Greatest integer function
$[x] = n where n \leq x < n + 1$
range Z
GIF bounds
$x - 1 < [x] \leq x$
[x]+[-x]
$[x] + [- x] = {0 - 1 x \in Z x \in / Z$
Q1MCQ[x]+[-x]
For a non-integer $x$, $[x]+[-x]$ equals:
- A
  $-1$
- B
  $0$
- C
  $1$
- D
  $2x$
Fractional part
${x} = x - [x], 0 \leq {x} < 1$
range [0,1)
Signum function
$sgn (x) = ⎩ ⎨ ⎧ 10 - 1 x > 0 x = 0 x < 0$
Range of ax²+bx+c (a>0)
$[\frac{4 a c - b ^{2}}{4 a}, \infty)$
Q1MCQRange of quadratic
The range of $f(x)=x^{2}+2x+3$ is:
- A
  $[2,\infty)$
- B
  $[3,\infty)$
- C
  $(-\infty,2]$
- D
  $\mathbb{R}$
Domain of √(a²−x²)
$[- a, a]$
Q1MCQDomain √(a²−x²)
The domain of $\sqrt{9-x^{2}}$ is:
- A
  $[-3,3]$
- B
  $(-3,3)$
- C
  $[0,3]$
- D
  $\mathbb{R}$
Domain of √(x²−a²)
$(- \infty, - a] \cup [a, \infty)$
Q1MCQDomain √(x²−a²)
The domain of $\sqrt{x^{2}-4}$ is:
- A
  $(-\infty,-2]\cup[2,\infty)$
- B
  $[-2,2]$
- C
  $[2,\infty)$
- D
  $\mathbb{R}$
Domain of logₐx
$(0, \infty)$
Q1MCQDomain of log
The domain of $\log x$ is:
- A
  $(0,\infty)$
- B
  $[0,\infty)$
- C
  $\mathbb{R}$
- D
  $(-\infty,0)$
Periodic function
$f (x + T) = f (x) \forall x$
T = fundamental period
Q1MCQPeriodic
A function with $f(x+T)=f(x)$ for the least positive $T$ is:
- A
  periodic with period $T$
- B
  even
- C
  odd
- D
  one-one
Period of sinⁿx / cosⁿx
$2 π (n odd), π (n even)$
Q1MCQPeriod sinⁿ
The period of $\sin^{2}x$ is:
- A
  $\pi$
- B
  $2\pi$
- C
  $\dfrac{\pi}{2}$
- D
  $4\pi$
Period of |sin x| etc.
$π$
Q1MCQPeriod |sin x|
The period of $|\sin x|$ is:
- A
  $\pi$
- B
  $2\pi$
- C
  $\dfrac{\pi}{2}$
- D
  $4\pi$
Period of f(ax+b)
$\frac{T}{∣ a ∣}$
T = period of f
Q1MCQPeriod f(ax+b)
The period of $\sin(2x)$ is:
- A
  $\pi$
- B
  $2\pi$
- C
  $\dfrac{\pi}{2}$
- D
  $4\pi$
LCM rule for period
$T = LCM (T_{1}, T_{2})$
sum of periodic functions
Q1MCQLCM period
The period of $\sin x+\cos x$ is:
- A
  $2\pi$
- B
  $\pi$
- C
  $\dfrac{\pi}{2}$
- D
  $4\pi$
Functional eqn f(x+y)=f(x)f(y)
$f (x) = k^{x}$
Q1MCQf(x+y)=f(x)f(y)
If $f(x+y)=f(x)f(y)$ for all $x,y$, then $f(x)$ is of the form:
- A
  $k^{x}$
- B
  $kx$
- C
  $k\log x$
- D
  $1+x^{n}$
Functional eqn f(xy)=f(x)+f(y)
$f (x) = k lo g_{a} x$
Q1MCQf(xy)=f(x)+f(y)
If $f(xy)=f(x)+f(y)$, then $f(x)$ is of the form:
- A
  $k\log_a x$
- B
  $k^{x}$
- C
  $kx$
- D
  $1\pm x^{n}$
Functional eqn f(x)+f(1/x)=f(x)f(1/x)
$f (x) = 1 \pm x^{n}$
Q1MCQf(x)+f(1/x)=f(x)f(1/x)
A polynomial with $f(x)+f\!\left(\tfrac1x\right)=f(x)f\!\left(\tfrac1x\right)$ is of the form:
- A
  $1\pm x^{n}$
- B
  $k^{x}$
- C
  $k\log x$
- D
  $kx$
Shift f(x)+a and f(x+a)
$f (x) + a : vertical shift; f (x + a) : horizontal shift (left if a > 0)$
Q1MCQGraph shift
The graph of $y=f(x+2)$ is the graph of $y=f(x)$ shifted:
- A
  $2$ units left
- B
  $2$ units right
- C
  $2$ units up
- D
  $2$ units down
Graph of y=|f(x)|
$reflect the below-axis part of f above the x-axis$
Q1MCQGraph of |f(x)|
To draw $y=|f(x)|$ from $y=f(x)$:
- A
  reflect the part below the x-axis above it
- B
  reflect the whole graph about the y-axis
- C
  shift up by $1$
- D
  reflect about $y=x$

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Function formulas

Function, every formula

Number of functions A→B

One-one (injection)

Number of injections A→B

Onto (surjection) condition

Number of onto functions (n(B)=2)

Bijection: n(A)=n(B)

Number of bijections

Composite function

Composite not commutative

Inverse of a composite

Inverse function condition

Inverse pair identities

Graph of f and f⁻¹

Even function

Odd function

Even + odd decomposition

Modulus function

Greatest integer function

GIF bounds

[x]+[-x]

Fractional part

Signum function

Range of ax²+bx+c (a>0)

Domain of √(a²−x²)

Domain of √(x²−a²)

Domain of logₐx

Periodic function

Period of sinⁿx / cosⁿx

Period of |sin x| etc.

Period of f(ax+b)

LCM rule for period

Functional eqn f(x+y)=f(x)f(y)

Functional eqn f(xy)=f(x)+f(y)

Functional eqn f(x)+f(1/x)=f(x)f(1/x)

Shift f(x)+a and f(x+a)

Graph of y=|f(x)|

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