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MATHEMATICS > FUNCTIONS > Flashcards

FUNCTIONS Flashcards

1
Q

WHAT IS A FUNCTION

A 
f : A- → B relation is said to be
-
a
function if every
element of A
is mapped with unique
element in B .
i.e every
element in A has only one
image in B .
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2
Q

DESCRIBE A FUNCTION IN TERMS OF 1-1 OR MANY-1 1-MANY AND MANY-MANY

A 
Every function is either I - l of many
one
relation .
But converse
is not true
i -e every l - I
,
many one
relation need not be
function .
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3
Q

WHAT IS DOMAIN

A

THE COLLECTION OF ALL INITIAL ELEMENTS

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4
Q

WHAT IS RANGE

A

COLLECTION OF IMAGES OF DOMAIN

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5
Q

WHAT IS CODOMAIN

A

COLLECTION OF FINAL RESULTS. RANGE IS A SUBSET OF THIS

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6
Q

WHAT IS IMAGE

A

F(D)

D IS ANY ELEMENT IN DOMAIN

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7
Q

WHAT IS PREIMAGE

A

OPPOSITE OF IMAGE

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8
Q

UNDER WHAT CONDITION CAN YOU. USE OPERATION OM FUNCTIONS

A

YOU CAN ONLY USE IT ON. THE INTERSECTION OF 2 DOMAINS

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9
Q 
Domain of f= {1,2)  (3,-2)  (5,6)  (0,4)
Domain of g= {1,8)  (2,1). (5,0),   (4,1)}
find f+g
f-g
f*g
f/g
A

f+g={{1,10],[5,6]}
f-g={ ( 1,-6) ,( 5,6) }
f*g={ (1,16)( 5,0) }
f/g={1,2/8} (REMEMBER THAT YOU DONT CONSIDER 5 AS DIVISION BY 0 IS NOT POSSIBLE)

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10
Q

domain of root(f)

A

x/f(x)>0

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11
Q

NUMBER OF FUNCTIONS

A

IF n(A) =m. n (B) = n
THEN
n^m

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12
Q

WHAT IS 1-1 OR INJECTIVE FUNCTION

A

if all distinct elements in A

have distinct images in B .

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13
Q

what are the characteristics of an injective function

A 
If f is 1 - 1 function then no two
elements in A have same image
in B .
( ii) every
element in range
has exactly
one pre image in domain
( iii) every
element in codomain
has
at most one pre image in domain
(iv) n(A)<=n(B)
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14
Q

how do you check a function using graphs

A

A- graph represent function if every
vertical line from domain intersects graph in
exactly one point .

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15
Q

how do you check a injective function using graphs

A

If every Horizontal line
from codomain intersect graph of f
in almost one point(0 or 1 point)

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16
Q

what can you say about a continuous graph

A

A continuous function graph is 1-1
if graph is strictly increasing or
strictly decreasing .

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17
Q

NUMBER OF INJECTIVE FUNCTIONS

A

nPm

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18
Q

what is many one function

A 
function
is said to be many one function
if f is not 1-1
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19
Q

NUMBER OF MANY ONE FUNCTIONS

A

n^m-nPm

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20
Q

what happens if m>n

A

number of many one functions=n^m

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21
Q

what is onto function(surjective function)

A 
function is said to be
onto function if every
element of
codomain has atleast one preimage in
domain . 
range=codoman
n(A)>n(B)
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22
Q

how do you check a surjective function using graphs

A

graph represents
onto ( surjective ) function if every
Horizontal line from codomain intersect
graph in atleast one point

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23
Q

number of onto functions

A

n^m-nC1(n-m)^m+nC2(n-2)^m-nC3(n-3)^m………

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24
Q

what is bijective

A

function is
said to be bijective if f is 1-1 and onto
every

element in codomain has exactly

one pre image .
n(A)=n(B)

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25
Q

number of bijective functions

A

n!

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26
Q

how do you check a bijective function using graphs

A

function graph is bijective

if every

horizontal line from codomain intersect

graph in exactly one point .

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27
Q

what is constant function

A

said to be

-
constant function if all elements in A
are mapping with same

element in B .
ie if Range is singleton set then that
function is called as

constant function .

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28
Q

number of constant functions

A

n

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29
Q

how do you check a constant function using graphs

A

then

graph of f is parallel to x-axis.

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30
Q

what is identity function

A

function is said to be identity function if f(x)=x

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31
Q

graph of exponential function

A

2 cases
0<a>1
SEE GRAPHS</a>

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32
Q

graph of logarithmic function

A

2 cases
0<a>1
SEE GRAPHS</a>

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33
Q

what is modulus function

A

f(x)=x. x>0
=0 x=0
=-x x<0

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34
Q

domain and range of modulus function

A

R

0.infinity

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35
Q

root(x^2)=?

A

|x|

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36
Q

if |x|=k

A

x=+-k if k>0

null set if k<0

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37
Q

if |x|

A

x belongs to(-k,k)

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38
Q

if |x|>k

A

x belongs to (-infinity,-k)U(k,infinity)

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39
Q

|x+y|<=?

A

|x|+|y|

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40
Q

|x-y|>=

A

||x|-|y||

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41
Q

|x^2-5x+6|=1/8 has how many solutions

A

DRAW GRAPH AND DO

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42
Q

graph of MODULUS function

A

SEE GRAPH

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43
Q

graph of GREATEST INTEGER function

A

SEE GRAPH

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44
Q

|XY|=?

A

|X||Y|

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45
Q

|X/Y|

A

|X|/|Y|

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46
Q

equalilties of [x]

A

[x]

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47
Q

[x]+[-x]=?

A

0

-1

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48
Q

[-x]

A
  • [x]

- 1-[x]

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49
Q

[x+y]>=?

A

[x]+[y]

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50
Q

If p is a prime number then

exponent of P in n !

A

[n/p]+[n/p^2]…….

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51
Q

number of zeroes ending in n!

A

if p is 5

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52
Q

express [x] in terms of fractions

A

[x/2]+[(x+1)/2]

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53
Q

[x+n]=?

A

[x]+n

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54
Q

what is fractional part

A

{x}=x-{x}

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55
Q

{x+n}=?

A

{x}

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56
Q

graph of {x} function

A

SEE GRAPH

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57
Q

{x+y}<=

A

{x}+{y}

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58
Q

{x}+{-x}

A

1

0

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59
Q

graph of -{x} function

A

SEE GRAPH

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60
Q

opposite of onto function

A

into function

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61
Q

what is signum function

A

f(x)=1 x>0
f(x)=0 x=0
f(x)=-1 x<0

62
Q

what are identical function

A

2 functions are identical if domain of f and g are same and f(x)=g(x)

63
Q

for what interval is 2lox and logx^2 identical

A

(0,infinity)

64
Q

what is an inverse function

A 
if f(x)=y then
inverse function is f(y)=x
65
Q

for what functions are inverse functions defined

A

ONLY BIJECTIVE FUNCTIONS

66
Q

domain of f=

range of f=

A

range of f^-1

domain. of f^-1

67
Q

if a,b is a point then what is f^-1(b)

A

a

68
Q

how to represent inverse functions in a graph

A

y=f^-1(x) is a image of y=f(x) in x=y line

69
Q

where do y=f^-1(x) and y=f(x) intersect

A

either x=y line or x=-y line

70
Q

f(x)=3x+4 find y=f^-1(x)

A

(x-4)/3

71
Q

Find inverse function of 2^x

A

log_2. x

72
Q

are inverse trigonometric functions bijective

A

no

but from -pi/2 to pi/2 YES

73
Q

draw sin^-1 x graph

A

SEE GRAPH

74
Q

draw cos^-1 x graph

A

SEE GRAPH

75
Q

what happens if f(x) is increasing

A

then f^-1(x) increases too

76
Q

what happens if f(x) is decreasing

A

then f^-1(x) decreases too

77
Q

if y=f(x) is concave up

A

then y=f^-1(x) is concave down

78
Q

if y=f(x) is concave down

A

then y=f^-1(x) is concave up

79
Q

how do we get the graph of y=f^-1(x)

A

by rotating y=f(x) graph anticlockwise direction with 90 degrees and take image of this in y axis
OR
by rotating it clockwise by 90 degrees and taking image of this in x axis

80
Q

what are the asymptotes for y=cot^-1x

A

y=0

y=pi

81
Q

what is asymptote of sec^-1x

A

y=pi/2

82
Q

what is asymptote of cosec^-1x

A

x axis

83
Q

sin^-1+cos^-1=?

A

pi/2

84
Q

tan^-1+cot^-1=?

A

pi/2

85
Q

cosec^-1+sec^-1=?

A

pi/2

86
Q 
draw graphs of sin^-1
cos^-1
tan^-1
cot^-1
cosec^-1
sec^-1
A

SEE GRAPHS

87
Q

if f(x) is quadratic equation then what is its range?

A

if a>0 then it’s (4ac-b^2)/4a to infinity

if a<0 then it’s -infinity to 4ac-b^2)/4a

88
Q

if f’(x)>0 then

A

y=f(x) is increasing

89
Q

if f’(x)<0 then

A

y=f(x) is decreasing

90
Q

range of odd degree polynomial

A

R

91
Q

where is a polynomial increasing and decreasing

A

first diffrentiate the equation and equate it to 0
plot the values of x in the graph like wavy curve
then the graph is increasing in the positive areas of the wavy curve and it is decreasing in the negative areas of the wavy curve
these values of x are the local minima and maxima of the polynomial

92
Q

if f’(x)>0 and f’(x)=0 is possible at discrete points

A

y=f(x) is strictly increasing

93
Q

If f(x) =0 continuously in some internal

A

y=f(x) is not 1-1

94
Q

if f’(x)>0

A

it is 1-1

95
Q

if f’(x)>=0

A

it is 1-1 if it is 0 at discrete points

96
Q

if f’(x)<0

A

it is 1-1

97
Q

if f’(x)<=0

A

it is 1-1 if it is 0 at discrete points

98
Q 
If f(x) is even degree polynomial with
leading coefficient positive
A

then range is (m,infinity) where m is any integer

99
Q 
If f(x) is even degree polynomial with
leading coefficient negative
A

then range is (-infinty,m) where m is any integer

100
Q

how to find the value of m

A

diffrentiate it and use the wavy curve method

101
Q

If f(x) is even degree polynomial

A

then it is not onto and not 1-1

102
Q

range of asinx+bcosx+c

A

c-root(a^2+b^2) to c+root(a^2+b^2)

103
Q

range using am gm

A

(x+y)/2>=root(xy)

104
Q

when is a function always self inverse function

A

(ax+b)/(cx-a)

105
Q

range of f(x)=(x-@)(ax+b)/(x-a)

A

R-(a@+b)

106
Q

range of f(x)=(x-@)/(x-@)(ax+b)

A

R-{0,1/a@+b)

107
Q

range of f(x)=(x-@)(ax+b)/(x-@)(cx+d)

A

R-{a@=b/c@+d,a/c}

108
Q

if f(x) is a expression with its numerator and denominator containing polynomials without a common factor

A

then first assume the expression to be equal to y. then form a quadratic equation in x.
use D>0 and find the range of y

109
Q

ax^2+bx+c/px^2+qx+r.

and no common factor for numerator and denominator then

A

it is always many-one.

110
Q

what is an even function

A

if f(x)=f(-x)

111
Q

what is an example of an even function

A

polynomials with even degrees

cos function

112
Q

even function is always symmetric about

A

y axis

113
Q

if f(x) is an even function

A

then k.f(x) (f(x))^n root(f(x)) log(f(x)) e^f(x) a^f(x) are all even functions

114
Q

if f(x) is any function then f(x)+f(-x) is

A

always even function

115
Q

what is an odd function

A

f(-x)=f(x)

116
Q

what are examples of odd functions

A

sinx
tanx
polynomials with only odd exponents

117
Q

if 0 is in the domain of odd functions

A

f(0)=0

118
Q

graph of odd function is always symmetric about

A

origin

119
Q

if (a,b) is a point on f(x) then what point also lies on this function

A

(-a,-b)

120
Q

if f(x) is an odd function then (f(x))^n

A

is odd if n is odd

and is even if n is even

121
Q

what is the only function which is odd and even

A

f(x)=0

122
Q

if f(x) is any function then f(x)-f(-x) is ?

A

odd function

123
Q

every function can be expressed as

A

sum of odd and even function

124
Q

if y=f(x) is a diffrential function

A 
if f(x) is odd then f'(x) is even 
if f(x)is even then f'(x) is odd
125
Q

if f(x) and g(x) are even functions then

A 
f(x)+g(x)
f(x)-g(x)
f(x).g(x)
f(x)/g(x)
are always even in their domain
126
Q

if f(x) and g(x) are odd functions then

A 
f(x)+g(x)
f(x)-g(x) are odd functions
while f(x).g(x)
f(x)/g(x) are even functions
127
Q

f(x) is odd and g(x) is even then what is neither odd nor even

A

f(x)+g(x)

f(x)-g(x)

128
Q

-x+root(x^2+1)=

A

1/(x+root(x^2+1))

129
Q

if you have to prove if a function is even and there are complex terms in the form of fractions then what do you do

A

take the fucking LCM

130
Q

what is even extension

A

f[a,b] belongs to R where ab>0 then the even extensionof f is g where g:[-b,-a] belongs to R SUCH THAT g(x)=f(-x)

131
Q

what is odd extension

A

if[a,b] belongs to R where ab>0 then the even extensionof f is g where g:[-b,-a] belongs to R SUCH THAT g(x)=-f(-x)

132
Q

if f(x) is symmetric about x=a

A

f(x)=f(2a-x)

f(a-x)=f(a+x)

133
Q

if a polynomial of degree 4 has only three roots then what is the sum of the roots

A

you have to include the repeated root twice

134
Q

if f(x) is symmetric about a point (a,0)

A 
then f(a-x)=-f(a+x)
f(x)=-f(2a+x)
135
Q

every odd function is symmetric about

A

(0,0)

136
Q

how to solve questions like f(x)=f(x+1/x+2) where f(x) is an even function

A

equate x=x+1/x+2

and -x=x+1/x+2

137
Q

how to solve questions like if y=f(x) is symmetric about x=2 line then find the value of x satisfying f(x)=f(x+1/x+2)

A

equate x and 4-x to the respective equation

138
Q

while solving problems before choosing neither even nor odd what should you do

A

substitute some values and check if it is satisfying or not and then choose the option correctly

139
Q

what is a periodic function?

A

if there exists a positive real number T such that f(x+T)=f(x)

140
Q

what is fundamental period

A

the smallest value of T is called fundamental period

141
Q

what is the fundamental period of a constant function

A

a constant function is a periodic function but its period is not defined

142
Q

if f(x) is periodic with period T then

A

2T,3T,nT are also periods of f(x)

143
Q

if f(x) is periodic with period T then

A

f(x)+k, f(x)-k, f(x+k), f(x-k), kf(x) 1/k*f(x), kf(x)+l, kf(x+v)+l. log(f(x). e/a^f(x) are all periodic with period T

144
Q

if f(x) is periodic with period T then what is the period of f(ax+b)

A

T/|a|

145
Q

if f(x) is periodic with period T then

A

(f(x))^n, (f(x))^1/n root(f(x)) [f(x)] |f(x)| {f(x)} and g(f(x)) where g is any function then T is the period but it may not be the fundamental period

146
Q

if f(x) is periodic with period T then

A

f(x^n),f( root(x) are not periodic

147
Q

lcm of fractions

A

lcm of numerators/hcf of denominators

148
Q

if y=f(x) is periodic with period T1 and y=g(x) is periodic with period T2

A

then f(x)+g(x). f(x)-g(x). f(x)/g(x). f(x)/g(x). k(f(x)+l(g(x) are periodic with period as lcm of T1 and T2

149
Q

if f(x+T)+f(x)=k

A

then f is a periodic function with period 2T

150
Q

if f(x+a)+f(x+b)=k

A

then period is 2|b-a|

151
Q

y = f(x) is symmetric about x=a

and x=b lines

A

period is 2|b-a|

MATHEMATICS (2 decks)

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