FUNCTIONS Flashcards
WHAT IS A FUNCTION
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 .
DESCRIBE A FUNCTION IN TERMS OF 1-1 OR MANY-1 1-MANY AND MANY-MANY
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 .
WHAT IS DOMAIN
THE COLLECTION OF ALL INITIAL ELEMENTS
WHAT IS RANGE
COLLECTION OF IMAGES OF DOMAIN
WHAT IS CODOMAIN
COLLECTION OF FINAL RESULTS. RANGE IS A SUBSET OF THIS
WHAT IS IMAGE
F(D)
D IS ANY ELEMENT IN DOMAIN
WHAT IS PREIMAGE
OPPOSITE OF IMAGE
UNDER WHAT CONDITION CAN YOU. USE OPERATION OM FUNCTIONS
YOU CAN ONLY USE IT ON. THE INTERSECTION OF 2 DOMAINS
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/gf+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)
domain of root(f)
x/f(x)>0
NUMBER OF FUNCTIONS
IF n(A) =m. n (B) = n
THEN
n^m
WHAT IS 1-1 OR INJECTIVE FUNCTION
if all distinct elements in A
have distinct images in B .
what are the characteristics of an injective function
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)
how do you check a function using graphs
A- graph represent function if every
vertical line from domain intersects graph in
exactly one point .
how do you check a injective function using graphs
If every Horizontal line
from codomain intersect graph of f
in almost one point(0 or 1 point)
what can you say about a continuous graph
A continuous function graph is 1-1
if graph is strictly increasing or
strictly decreasing .
NUMBER OF INJECTIVE FUNCTIONS
nPm
what is many one function
function is said to be many one function if f is not 1-1
NUMBER OF MANY ONE FUNCTIONS
n^m-nPm
what happens if m>n
number of many one functions=n^m
what is onto function(surjective function)
function is said to be onto function if every element of codomain has atleast one preimage in domain . range=codoman n(A)>n(B)
how do you check a surjective function using graphs
graph represents
onto ( surjective ) function if every
Horizontal line from codomain intersect
graph in atleast one point
number of onto functions
n^m-nC1(n-m)^m+nC2(n-2)^m-nC3(n-3)^m………
what is bijective
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)
number of bijective functions
n!
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 .
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 .
number of constant functions
n
how do you check a constant function using graphs
then
graph of f is parallel to x-axis.
what is identity function
function is said to be identity function if f(x)=x
graph of exponential function
2 cases
0<a>1
SEE GRAPHS</a>
graph of logarithmic function
2 cases
0<a>1
SEE GRAPHS</a>
what is modulus function
f(x)=x. x>0
=0 x=0
=-x x<0
domain and range of modulus function
R
0.infinity
root(x^2)=?
|x|
if |x|=k
x=+-k if k>0
null set if k<0
if |x|
x belongs to(-k,k)
if |x|>k
x belongs to (-infinity,-k)U(k,infinity)
|x+y|<=?
|x|+|y|
|x-y|>=
||x|-|y||
|x^2-5x+6|=1/8 has how many solutions
DRAW GRAPH AND DO
graph of MODULUS function
SEE GRAPH
graph of GREATEST INTEGER function
SEE GRAPH
|XY|=?
|X||Y|
|X/Y|
|X|/|Y|
equalilties of [x]
[x]
[x]+[-x]=?
0
-1
[-x]
- [x]
- 1-[x]
[x+y]>=?
[x]+[y]
If p is a prime number then
exponent of P in n !
[n/p]+[n/p^2]…….
number of zeroes ending in n!
if p is 5
express [x] in terms of fractions
[x/2]+[(x+1)/2]
[x+n]=?
[x]+n
what is fractional part
{x}=x-{x}
{x+n}=?
{x}
graph of {x} function
SEE GRAPH
{x+y}<=
{x}+{y}
{x}+{-x}
1
0
graph of -{x} function
SEE GRAPH
opposite of onto function
into function
what is signum function
f(x)=1 x>0
f(x)=0 x=0
f(x)=-1 x<0
what are identical function
2 functions are identical if domain of f and g are same and f(x)=g(x)
for what interval is 2lox and logx^2 identical
(0,infinity)
what is an inverse function
if f(x)=y then inverse function is f(y)=x
for what functions are inverse functions defined
ONLY BIJECTIVE FUNCTIONS
domain of f=
range of f=
range of f^-1
domain. of f^-1
if a,b is a point then what is f^-1(b)
a
how to represent inverse functions in a graph
y=f^-1(x) is a image of y=f(x) in x=y line
where do y=f^-1(x) and y=f(x) intersect
either x=y line or x=-y line
f(x)=3x+4 find y=f^-1(x)
(x-4)/3
Find inverse function of 2^x
log_2. x
are inverse trigonometric functions bijective
no
but from -pi/2 to pi/2 YES
draw sin^-1 x graph
SEE GRAPH
draw cos^-1 x graph
SEE GRAPH
what happens if f(x) is increasing
then f^-1(x) increases too
what happens if f(x) is decreasing
then f^-1(x) decreases too
if y=f(x) is concave up
then y=f^-1(x) is concave down
if y=f(x) is concave down
then y=f^-1(x) is concave up
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
what are the asymptotes for y=cot^-1x
y=0
y=pi
what is asymptote of sec^-1x
y=pi/2
what is asymptote of cosec^-1x
x axis
sin^-1+cos^-1=?
pi/2
tan^-1+cot^-1=?
pi/2
cosec^-1+sec^-1=?
pi/2
draw graphs of sin^-1 cos^-1 tan^-1 cot^-1 cosec^-1 sec^-1
SEE GRAPHS
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
if f’(x)>0 then
y=f(x) is increasing
if f’(x)<0 then
y=f(x) is decreasing
range of odd degree polynomial
R
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
if f’(x)>0 and f’(x)=0 is possible at discrete points
y=f(x) is strictly increasing
If f(x) =0 continuously in some internal
y=f(x) is not 1-1
if f’(x)>0
it is 1-1
if f’(x)>=0
it is 1-1 if it is 0 at discrete points
if f’(x)<0
it is 1-1
if f’(x)<=0
it is 1-1 if it is 0 at discrete points
If f(x) is even degree polynomial with leading coefficient positive
then range is (m,infinity) where m is any integer
If f(x) is even degree polynomial with leading coefficient negative
then range is (-infinty,m) where m is any integer
how to find the value of m
diffrentiate it and use the wavy curve method
If f(x) is even degree polynomial
then it is not onto and not 1-1
range of asinx+bcosx+c
c-root(a^2+b^2) to c+root(a^2+b^2)
range using am gm
(x+y)/2>=root(xy)
when is a function always self inverse function
(ax+b)/(cx-a)
range of f(x)=(x-@)(ax+b)/(x-a)
R-(a@+b)
range of f(x)=(x-@)/(x-@)(ax+b)
R-{0,1/a@+b)
range of f(x)=(x-@)(ax+b)/(x-@)(cx+d)
R-{a@=b/c@+d,a/c}
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
ax^2+bx+c/px^2+qx+r.
and no common factor for numerator and denominator then
it is always many-one.
what is an even function
if f(x)=f(-x)
what is an example of an even function
polynomials with even degrees
cos function
even function is always symmetric about
y axis
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
if f(x) is any function then f(x)+f(-x) is
always even function
what is an odd function
f(-x)=f(x)
what are examples of odd functions
sinx
tanx
polynomials with only odd exponents
if 0 is in the domain of odd functions
f(0)=0
graph of odd function is always symmetric about
origin
if (a,b) is a point on f(x) then what point also lies on this function
(-a,-b)
if f(x) is an odd function then (f(x))^n
is odd if n is odd
and is even if n is even
what is the only function which is odd and even
f(x)=0
if f(x) is any function then f(x)-f(-x) is ?
odd function
every function can be expressed as
sum of odd and even function
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
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
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
f(x) is odd and g(x) is even then what is neither odd nor even
f(x)+g(x)
f(x)-g(x)
-x+root(x^2+1)=
1/(x+root(x^2+1))
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
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)
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)
if f(x) is symmetric about x=a
f(x)=f(2a-x)
f(a-x)=f(a+x)
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
if f(x) is symmetric about a point (a,0)
then f(a-x)=-f(a+x) f(x)=-f(2a+x)
every odd function is symmetric about
(0,0)
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
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
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
what is a periodic function?
if there exists a positive real number T such that f(x+T)=f(x)
what is fundamental period
the smallest value of T is called fundamental period
what is the fundamental period of a constant function
a constant function is a periodic function but its period is not defined
if f(x) is periodic with period T then
2T,3T,nT are also periods of f(x)
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
if f(x) is periodic with period T then what is the period of f(ax+b)
T/|a|
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
if f(x) is periodic with period T then
f(x^n),f( root(x) are not periodic
lcm of fractions
lcm of numerators/hcf of denominators
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
if f(x+T)+f(x)=k
then f is a periodic function with period 2T
if f(x+a)+f(x+b)=k
then period is 2|b-a|
y = f(x) is symmetric about x=a
and x=b lines
period is 2|b-a|
