In 44-55 indicate which of the function in the referenced exercise are one-to-one correspondences. For each function that is a one-to-one correspondence, find the inverse function. Exercise 12b

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:

G:ℝ→ℝ is defined by the rule G(x)=2−3x, for all real numbers x.

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:

Before find its inverse first prove that G is both one-to-one and onto.

To prove G is one-to-one.

Let x1 and x2 be any two elements of ℝ.

Consider,

G(x1)=G(x2)      [We must show x1=x2]

By definition of G,

2−3x1=2−3x2

Subtracting 2 to both sides,

−3x1=−3x2

Divided by −3 from both side, gives:

x1=x2

Hence, G is one-to-one.

To proof G is onto

Let y∈ℝ