FUNCTIONS Flashcards
(151 cards)
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 .
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 .
3
Q
WHAT IS DOMAIN
A
THE COLLECTION OF ALL INITIAL ELEMENTS
4
Q
WHAT IS RANGE
A
COLLECTION OF IMAGES OF DOMAIN
5
Q
WHAT IS CODOMAIN
A
COLLECTION OF FINAL RESULTS. RANGE IS A SUBSET OF THIS
6
Q
WHAT IS IMAGE
A
F(D)
D IS ANY ELEMENT IN DOMAIN
7
Q
WHAT IS PREIMAGE
A
OPPOSITE OF IMAGE
8
Q
UNDER WHAT CONDITION CAN YOU. USE OPERATION OM FUNCTIONS
A
YOU CAN ONLY USE IT ON. THE INTERSECTION OF 2 DOMAINS
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)
10
Q
domain of root(f)
A
x/f(x)>0
11
Q
NUMBER OF FUNCTIONS
A
IF n(A) =m. n (B) = n
THEN
n^m
12
Q
WHAT IS 1-1 OR INJECTIVE FUNCTION
A
if all distinct elements in A
have distinct images in B .
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)
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 .
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)
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 .
17
Q
NUMBER OF INJECTIVE FUNCTIONS
A
nPm
18
Q
what is many one function
A
function is said to be many one function if f is not 1-1
19
Q
NUMBER OF MANY ONE FUNCTIONS
A
n^m-nPm
20
Q
what happens if m>n
A
number of many one functions=n^m
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)
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
23
Q
number of onto functions
A
n^m-nC1(n-m)^m+nC2(n-2)^m-nC3(n-3)^m………
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)
25
number of bijective functions
n!
26
how do you check a bijective function using graphs
function graph is bijective
=
if every
horizontal line from codomain intersect
graph in exactly one point .
27
what is constant function
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 .
28
number of constant functions
n
29
how do you check a constant function using graphs
then
graph of f is parallel to x-axis.
30
what is identity function
function is said to be identity function if f(x)=x
33
what is modulus function
f(x)=x. x>0
=0 x=0
=-x x<0
34
domain and range of modulus function
R
| 0.infinity
35
root(x^2)=?
|x|
36
if |x|=k
x=+-k if k>0
| null set if k<0
37
if |x|
x belongs to(-k,k)
38
if |x|>k
x belongs to (-infinity,-k)U(k,infinity)
39
|x+y|<=?
|x|+|y|
40
|x-y|>=
||x|-|y||
41
|x^2-5x+6|=1/8 has how many solutions
DRAW GRAPH AND DO
42
graph of MODULUS function
SEE GRAPH
43
graph of GREATEST INTEGER function
SEE GRAPH
44
|XY|=?
|X||Y|
45
|X/Y|
|X|/|Y|
46
equalilties of [x]
[x]
47
[x]+[-x]=?
0
| -1
48
[-x]
- [x]
| - 1-[x]
49
[x+y]>=?
[x]+[y]
50
If p is a prime number then
| exponent of P in n !
[n/p]+[n/p^2].......
51
number of zeroes ending in n!
if p is 5
52
express [x] in terms of fractions
[x/2]+[(x+1)/2]
53
[x+n]=?
[x]+n
54
what is fractional part
{x}=x-{x}
55
{x+n}=?
{x}
56
graph of {x} function
SEE GRAPH
57
{x+y}<=
{x}+{y}
58
{x}+{-x}
1
| 0
59
graph of -{x} function
SEE GRAPH
60
opposite of onto function
into function
61
what is signum function
f(x)=1 x>0
f(x)=0 x=0
f(x)=-1 x<0
62
what are identical function
2 functions are identical if domain of f and g are same and f(x)=g(x)
63
for what interval is 2lox and logx^2 identical
(0,infinity)
64
what is an inverse function
```
if f(x)=y then
inverse function is f(y)=x
```
65
for what functions are inverse functions defined
ONLY BIJECTIVE FUNCTIONS
66
domain of f=
| range of f=
range of f^-1
| domain. of f^-1
67
if a,b is a point then what is f^-1(b)
a
68
how to represent inverse functions in a graph
y=f^-1(x) is a image of y=f(x) in x=y line
69
where do y=f^-1(x) and y=f(x) intersect
either x=y line or x=-y line
70
f(x)=3x+4 find y=f^-1(x)
(x-4)/3
71
Find inverse function of 2^x
log_2. x
72
are inverse trigonometric functions bijective
no
| but from -pi/2 to pi/2 YES
73
draw sin^-1 x graph
SEE GRAPH
74
draw cos^-1 x graph
SEE GRAPH
75
what happens if f(x) is increasing
then f^-1(x) increases too
76
what happens if f(x) is decreasing
then f^-1(x) decreases too
77
if y=f(x) is concave up
then y=f^-1(x) is concave down
78
if y=f(x) is concave down
then y=f^-1(x) is concave up
79
how do we get the graph of y=f^-1(x)
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
what are the asymptotes for y=cot^-1x
y=0
| y=pi
81
what is asymptote of sec^-1x
y=pi/2
82
what is asymptote of cosec^-1x
x axis
83
sin^-1+cos^-1=?
pi/2
84
tan^-1+cot^-1=?
pi/2
85
cosec^-1+sec^-1=?
pi/2
86
```
draw graphs of sin^-1
cos^-1
tan^-1
cot^-1
cosec^-1
sec^-1
```
SEE GRAPHS
87
if f(x) is quadratic equation then what is its range?
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
if f'(x)>0 then
y=f(x) is increasing
89
if f'(x)<0 then
y=f(x) is decreasing
90
range of odd degree polynomial
R
91
where is a polynomial increasing and decreasing
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
if f'(x)>0 and f'(x)=0 is possible at discrete points
y=f(x) is strictly increasing
93
If f(x) =0 continuously in some internal
y=f(x) is not 1-1
94
if f'(x)>0
it is 1-1
95
if f'(x)>=0
it is 1-1 if it is 0 at discrete points
96
if f'(x)<0
it is 1-1
97
if f'(x)<=0
it is 1-1 if it is 0 at discrete points
98
```
If f(x) is even degree polynomial with
leading coefficient positive
```
then range is (m,infinity) where m is any integer
99
```
If f(x) is even degree polynomial with
leading coefficient negative
```
then range is (-infinty,m) where m is any integer
100
how to find the value of m
diffrentiate it and use the wavy curve method
101
If f(x) is even degree polynomial
then it is not onto and not 1-1
102
range of asinx+bcosx+c
c-root(a^2+b^2) to c+root(a^2+b^2)
103
range using am gm
(x+y)/2>=root(xy)
104
when is a function always self inverse function
(ax+b)/(cx-a)
105
range of f(x)=(x-@)(ax+b)/(x-a)
R-(a@+b)
106
range of f(x)=(x-@)/(x-@)(ax+b)
R-{0,1/a@+b)
107
range of f(x)=(x-@)(ax+b)/(x-@)(cx+d)
R-{a@=b/c@+d,a/c}
108
if f(x) is a expression with its numerator and denominator containing polynomials without a common factor
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
ax^2+bx+c/px^2+qx+r.
| and no common factor for numerator and denominator then
it is always many-one.
110
what is an even function
if f(x)=f(-x)
111
what is an example of an even function
polynomials with even degrees
| cos function
112
even function is always symmetric about
y axis
113
if f(x) is an even function
then k.f(x) (f(x))^n root(f(x)) log(f(x)) e^f(x) a^f(x) are all even functions
114
if f(x) is any function then f(x)+f(-x) is
always even function
115
what is an odd function
f(-x)=f(x)
116
what are examples of odd functions
sinx
tanx
polynomials with only odd exponents
117
if 0 is in the domain of odd functions
f(0)=0
118
graph of odd function is always symmetric about
origin
119
if (a,b) is a point on f(x) then what point also lies on this function
(-a,-b)
120
if f(x) is an odd function then (f(x))^n
is odd if n is odd
| and is even if n is even
121
what is the only function which is odd and even
f(x)=0
122
if f(x) is any function then f(x)-f(-x) is ?
odd function
123
every function can be expressed as
sum of odd and even function
124
if y=f(x) is a diffrential function
```
if f(x) is odd then f'(x) is even
if f(x)is even then f'(x) is odd
```
125
if f(x) and g(x) are even functions then
```
f(x)+g(x)
f(x)-g(x)
f(x).g(x)
f(x)/g(x)
are always even in their domain
```
126
if f(x) and g(x) are odd functions then
```
f(x)+g(x)
f(x)-g(x) are odd functions
while f(x).g(x)
f(x)/g(x) are even functions
```
127
f(x) is odd and g(x) is even then what is neither odd nor even
f(x)+g(x)
| f(x)-g(x)
128
-x+root(x^2+1)=
1/(x+root(x^2+1))
129
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
take the fucking LCM
130
what is even extension
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
what is odd extension
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
if f(x) is symmetric about x=a
f(x)=f(2a-x)
| f(a-x)=f(a+x)
133
if a polynomial of degree 4 has only three roots then what is the sum of the roots
you have to include the repeated root twice
134
if f(x) is symmetric about a point (a,0)
```
then f(a-x)=-f(a+x)
f(x)=-f(2a+x)
```
135
every odd function is symmetric about
(0,0)
136
how to solve questions like f(x)=f(x+1/x+2) where f(x) is an even function
equate x=x+1/x+2
| and -x=x+1/x+2
137
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)
equate x and 4-x to the respective equation
138
while solving problems before choosing neither even nor odd what should you do
substitute some values and check if it is satisfying or not and then choose the option correctly
139
what is a periodic function?
if there exists a positive real number T such that f(x+T)=f(x)
140
what is fundamental period
the smallest value of T is called fundamental period
141
what is the fundamental period of a constant function
a constant function is a periodic function but its period is not defined
142
if f(x) is periodic with period T then
2T,3T,nT are also periods of f(x)
143
if f(x) is periodic with period T then
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
if f(x) is periodic with period T then what is the period of f(ax+b)
T/|a|
145
if f(x) is periodic with period T then
(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
if f(x) is periodic with period T then
f(x^n),f( root(x) are not periodic
147
lcm of fractions
lcm of numerators/hcf of denominators
148
if y=f(x) is periodic with period T1 and y=g(x) is periodic with period T2
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
if f(x+T)+f(x)=k
then f is a periodic function with period 2T
150
if f(x+a)+f(x+b)=k
then period is 2|b-a|
151
y = f(x) is symmetric about x=a
| and x=b lines
period is 2|b-a|
