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 17 with the co-domain taken to be set of all real numbers not equal to 3

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)=3x−1x for all real numbers x≠0.

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−{0}→R−{3}, f(x)=3x−1x.

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)3x1−1x1=3x2−1x2x2(3x1−1)=x1(3x2−1)3x1x2−x2=3x1x2−x1−x2=−x1x2=x1

Hence, f is one-to-one