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 11b

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)=4x−5, 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:

Define G:ℝ→ℝ by the rule G(x)=4x−5, for all real numbers x

The objective is to determine one-to-one correspondence of the given function, if it is a one-to-one correspondence, find the inverse function.

Let us check if the function G(x) is one-to-one or not.

If x1 and x2 are any two elements of ℝ such that,

G(x1)=G(x2)

Now, by the definition of the function G(x) ,

4x1−5=4x2−5

Add 5 on both sides to get the following:

4x1=4x2

Divide both the sides by 4 to get the following:

x1=x2

Thus, the function G(x) is one-to-one function.

Let us check if the function G(x) is onto or not.

Suppose y is any real number, such that 4x−5=y

Then 4x=y+5

x=y+54∈R

Thus, every element from the range has a pre-image