Question

if f(x)=ax+b/bx+a, prove that f(x).f(1/x)=1

Answer 1

Given that

f(x) = (ax+b)/(bx+a)            ...(1)

Substitute x = 1/x in (1), we get

Then f(1/x) = (a/x+b)/(b/x+a)    

f(1/x) = [(a+bx)/x]/[(b+ax)/x]

f(1/x) = (a+bx)/(b+ax)

f(1/x) = (bx+a)/(ax+b)

Now

f(x)*f(1/x) =   [(ax+b)/(bx+a)]× [(bx+a)/(ax+b)]

f(x)*f(1/x) =  [(ax+b)(bx+a)]/[(ax+b)(bx+a)]

f(x)*f(1/x) =   1

Hence proved.

Answer 2

Answer:

Given that

f(x) = (ax+b)/(bx+a)

...(1)

Substitute x = 1 / x in (1), we get

Then f(1 / x) = (a / x + b) / (b / x + a)

f(1 / x) = [(a + bx) / x] / [(b + ax) / x]

f(1 / x) = (a + bx) / (b + ax) f(1 / x) = (bx + a) / (ax + b)

Now

f(x)*f(1/x) = [(ax+b)/(bx+a)]× [(bx+a)/(ax+b)]

f(x)*f(1/x) = [(ax+b)(bx+a)]/[(ax+b)(bx+a)]

f(x)*f(1/x) = 1

Hence proved.