Function formulas

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Formulas: 36
Subject: Mathematics
Exam: JEE Advanced
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Function, every formula

36 formulas, typeset and free. Print it, or keep it open beside your practice.

Number of functions A→B
$n (B)^{n (A)}$
One-one (injection)
$a_{1} \neq = a_{2} \Rightarrow f (a_{1}) \neq = f (a_{2})$
Number of injections A→B
$^{n} P_{r}, r = n (A), n = n (B), r \leq n$
Onto (surjection) condition
$f (A) = B (range = codomain)$
Number of onto functions (n(B)=2)
$2^{n} - 2, n = n (A)$
Bijection: n(A)=n(B)
$f bijection \Rightarrow n (A) = n (B)$
Number of bijections
$n! when n (A) = n (B) = n$
Composite function
$(g \circ f) (x) = g (f (x))$
Composite not commutative
$f \circ g \neq = g \circ f (in general)$
Inverse of a composite
$(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}$
Inverse function condition
$f invertible \Leftrightarrow f is a bijection$
Inverse pair identities
$f \circ f^{- 1} = I_{B}, f^{- 1} \circ f = I_{A}$
Graph of f and f⁻¹
$y = f (x) and y = f^{- 1} (x) are symmetric about y = x$
Even function
$f (- x) = f (x)$
symmetric about y-axis
Odd function
$f (- x) = - f (x)$
symmetric about origin
Even + odd decomposition
$f (x) = even \frac{1}{2} [f (x) + f (- x)] + odd \frac{1}{2} [f (x) - f (- x)]$
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]
$[x] + [- x] = {0 - 1 x \in Z x \in / Z$
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)
$[\frac{4 a c - b ^{2}}{4 a}, \infty)$
Domain of √(a²−x²)
$[- a, a]$
Domain of √(x²−a²)
$(- \infty, - a] \cup [a, \infty)$
Domain of logₐx
$(0, \infty)$
Periodic function
$f (x + T) = f (x) \forall x$
T = fundamental period
Period of sinⁿx / cosⁿx
$2 π (n odd), π (n even)$
Period of |sin x| etc.
$π$
Period of f(ax+b)
$\frac{T}{∣ a ∣}$
T = period of f
LCM rule for period
$T = LCM (T_{1}, T_{2})$
sum of periodic functions
Functional eqn f(x+y)=f(x)f(y)
$f (x) = k^{x}$
Functional eqn f(xy)=f(x)+f(y)
$f (x) = k lo g_{a} x$
Functional eqn f(x)+f(1/x)=f(x)f(1/x)
$f (x) = 1 \pm x^{n}$
Shift f(x)+a and f(x+a)
$f (x) + a : vertical shift; f (x + a) : horizontal shift (left if a > 0)$
Graph of y=|f(x)|
$reflect the below-axis part of f above the x-axis$

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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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