The function f 0 1 right arrow 0 infinity is defined by fx

The function f: (0, 1) right arrow (0, infinity) is defined by f(x) = x/l - x. Show that f is one-to-one and onto. Find the inverse function of f.

Solution

a) A function f from A to B is called onto if for all b in B there is an a in Asuch that f (a) = b.   All elements in B are used.

in the above problem A = {0,1} and B = {1,infinity)

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

Substituting 0 for x we get f(x) = 0/(1-0) = 0

Substituting 1 for x we get f(x) = 1/(1-1) = 1/0 = infinity

Here one value of x in A is always mapped to only one value in B. So it is one to one.

By definition above this is also onto

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

y = x/(1-x)

y(1-x) = x

y - yx = x

y = x+yx

y = x(1+y)

x(1+y) = y

x = y/(1+y)

the inverse of f(x) is

y = x/(1=X)

f\'(x) = x/(1+x)