Sets, Relations and Functions Test - 8

f(x) = 3.sin(√π² /16 −x² )
since the quantity within the square root can not be negative.
therefore
π² /16 −x² ≥0
x² ≤π² /16
−π/4 ≤x ≤π/4
the minimum value of the function is 3.sin0=0
and the maximum value of the function is 3sin π/4 = 3/√2
therefore range is [0, 3/√2]

 y = (x² - 3x + 2)/(x² + x -6)
= [(x-2)(x-1)]/[(x-2)(x+3)]
= (x-1)/(x+3)
= 1 - 4/(x+3)
x is not equal to 2 and -3
For x = 2, y = 1/5
for x = -3, y →1
Range (y): R - {1, 1/5}

f(x) = log[(1+x)/(1-x)]   ……….(1)
Substitute 2x/1+x² in place of x in equation(1)
f(2x1+x² )=log ⁡(1+2x1+x² )(1 −2x1+x² )
⟹f(2x/(1+x² ))=log[1+(2x/(1+x² ))/(1-(2x/1+x² ))]
⟹f(2x/(1+x² ))=log ⁡((1+x² +2x)/(1+x² −2x))
⟹f(2x/(1+x² ))=log((1+x)² / (1 −x))²
⟹f(2x/ (1+x² ))=log ⁡(1+x / 1 −x)²
We know that log ⁡mⁿ =nlog ⁡m
⟹f(2x/(1+x² ))=2log ⁡(1+x / 1 −x)
From equation (1):
f(2x/(1+x² ))=2f(x)

f(x) = (x-3)(x-1)/[(x2 -4)^1/2]
= x2 - 4 = 0
(x-2)(x+2) = 0
x = 2 and -2
(-∞, -2 ) U (2,∞)

For a function to have its inverse in a given domain, it should be continuous in that domain and should be a one-one function in that domain.
If the function is one-one in the domain, then it has to be strictly monotonic.
For example y=sin(x) has its domain in x ϵ[−π/2,π/2] since it is strictly monotonic and continuous in that domain.

Given that  n(A)=3 and  n(B)=4, the number of injections or one-one mapping is given by.

yx ​² + xy + y = x² - x + 1
(y-1)x² + (y+1)x + y-1 = 0
if y = -1
-2x² -1 -1 = 0
-2x² -2 = 0
Therefore, y cannot be equal to -1 since x will be a complex value
Now, y not equal to -1. 
(y-1)x² + (y+1)x + y-1 = 0  has real roots
hence, (y+1)² - 4 (y-1)(y-1) ≥0
y2 + 2y + 1 - 4y² + 8y - 4 ≥0
-3y² + 10y - 3 ≥0
3y² - 10y + 3 ≤0
3y² - 9y - y + 3 ≤0
3y (y-3) - 1 (y-3) ≤0
(3y-1)(y-3) ≤0
y ∈[1/3,3] 
Hence, range of function is [1/3,3]

f(x) = {x/2 x is even,   0 x is odd}
f(1) = f(3) = f(5) = f(odd) = 0
Hence f is not one - one. 
column = Z
Range = {0, even}
= {0,2,4,6,8,10.......}
f(x) = x/2 implies even
f(odd) = 0
Range is not equal to codomain  
Hence, onto

 f(x) = 3x - 4
Let f(x) = y
y = 3x - 4
y + 4 = 3x
x = (y+4)/3
x →f^-1 (x) and y →x
f-1(x) = (x+4)/3

The number of bijections from  A  to  A, will be  n! for  n  elements is  A.

Now there are  106, elements. 

Hence the number of bijections will be  n!=(106)!

aRb iff both a and b leave same remainder when divided by 5. 
a= 5x+n
a-1= 5x
b=5y+n
b-1 =5y
Equivalence class = {1,6,11,16,21}