In 44-55 indicate which of the functions in the referenced exercise are one-to-one correspondences. For each function that is a one-to-one correspondence, find the inverse function.

Exercise 18 with the co-domain taken to be the set of all real numbers not equal to 1

To check:

Whether the given function is one-to-one correspondence and also find the inverse of the given function if it exists.

Given information:

f(x)=x+1x−1 for all real numbers x≠1.

Concept used:

One-to-one function: The distinct elements in domain are mapped with distinct elements in co-domain.

Onto function: Every element in co-domain must be mapped with the element in domain.

Calculation:

Given that f:R−{1}→R−{1}, f(x)=x+1x−1

To show f is one-to-one correspondence, we need to show that f is one-to-one and onto.

To show f is one-to-one,

Let          f(x1)=f(x2)x1+1x1−1=x2+1x2−1(x2−1)(x1+1)=(x1−1)(x2+1)x2−x1=x1−x22x2=2x1x2=x1

Hence, f is one-to-one