funcns

1. If a(0) = x , a (n+1) = f( a ( n ) ) , n= 0 , 1 , 2 , ............................., find a(n) when

(i) f(x) = root of (mod x )

(ii) f(x) = 1 / 1-x

2. Let f(x) be a real valued funcn defined for all real nos x such that for some fixed real no. a>0 , f(x+a) = 1/2 + root of ( f(x) - square of f(x) ) and 1/2 less than equal to f(x) less than equal to 1 for all x . Show that f(x) is periodic with period '2a'.

3. Let f(x) = ( 2x^3 - 9x^2 + 12x +3) (x-a) /(x-a) . If range of f(x) is a proper subset of real nos. , then what is the most exhaustive set in which a lies??

3 Answers

39
Dr.House ·

2)

i dont know if i am being absolutely rubbish

but for f to have period 2a , we must have f(x)=f(x+2a)

putting x=x-a

f(x)=1/2+√[f(x-a)-√f(x-a)]

putting x=x+a

f(x+2a)=1/2+√[f(x+a)-√f(x+a)]

now these 2 are equal only when

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

so f is periodic with period 2a

39
Dr.House ·

1)

f(x)=1/(1-x)

a(0)=x and a(n+1)=f(a(n))

n=0 gives us a(1)=f(x)

so a(2)=f(a(1))=f(f(x))

and a(3)=f(f(f(x)))
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a(n)=f(f(f(.......f(x))))))))))) (n times f occurs)

a(1)=f(x)=1/1-x

a(2)=f(f(x))=f(1/(1-x))=(x-1)/x

a(3)=f(f(f(x)))=f((x-a)/x)=x

a(4) will similarly be 1/(1-x)

and series follows

1/(1-x) , (x-1)/x , x

these keep on occuring one after another

so a(n) willd epend on what remainder n leaves when divided by 3

if its 1 , then a(n)=1/(1-x)

if its 2, then a(n)=(x-1)/x

if its 0, then a(n)=x

39
Dr.House ·

3rd question , please look ta it again , i think u have made some mistake in typing the question

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