Show that the function f: 𝐑* →𝐑* defined by f(x)=(-)/(x) is one-one and onto, where 𝐑* is the se

Show that the function f: 𝐑* →𝐑* defined by f(x)=(-)/(x) is...

Key Concepts

Injectivity

Injectivity, or the one-to-one property, means that if two elements in the domain have the same image under the function, then they must be identical. In other words, a function f is injective if f(a) = f(b) implies a = b. This property is crucial when proving that a function is bijective, as it ensures that there is no overlap in the mapping from domain to codomain.

Surjectivity

Surjectivity, or the onto property, means that every element in the codomain is the image of some element from the domain. To show that a function is surjective, one must demonstrate that for every y in the codomain, there exists an x in the domain such that f(x) = y. This property is essential to ensure that the function covers the entire codomain.

Bijective Function

A bijective function is a function that is both injective and surjective. This means that every element of the codomain is mapped to by exactly one element from the domain. Bijective functions have the important property that they possess an inverse function, making them a central concept in many areas of mathematics, especially in algebra and analysis.

Domain and Codomain

The domain of a function is the set of possible inputs for the function, while the codomain is the set into which all outputs are constrained. The properties of a function such as injectivity and surjectivity depend heavily on the chosen domain and codomain. Changing either the domain or codomain can affect whether the function retains these properties.

Effects of Domain Change (from R* to N)

When the domain is changed from the set of all nonzero real numbers (R*) to the set of natural numbers (N), the behavior of the function can change dramatically. A function that is bijective on R* may no longer be surjective on N with the same codomain because natural numbers form a discrete and typically much smaller set compared to the entire collection of nonzero real numbers. This highlights the importance of considering the domain and codomain when examining the properties of functions.

Transcript

00:01 In this problem of relation and function, we have to show that the function f, which is from r dot to r dot, where r dot is the set of all non -zero real numbers, and the function is defined as fx is equals to 1 divide with x.

00:21 And now we have to show that this function is 1 -1 and on 2.

00:26 So first here, the function is defined from r dot to r dot where r dot is, the set of non -zero real numbers.

00:37 So now first we would check for 1 -1.

00:39 So check for 1 -1.

00:44 So when we check for 1 -1, so this implies that, say, f of x1, say this is equal to 1 divide with x1.

00:53 And now, f of, when we write f of x2, this is equal to 1 divide with x2.

01:00 If f of x1 is equal to f of x2, this is equal to f of x2, this is a implies that say if f of x1 is equals to f of x2 this implies that x1 should be equals to x2 now put the value so this would be x1 and this would have 1 divide with x2 so when we write it so do the cross multiplication from here we conclude that x2 is equal to x1 which is same as x1 is equal to x2 so hence we say that hence the function f is 1 1 1 and now now we have to check for onto function.

01:41 So we have given that fx is equals to 1 divided with x.

01:45 So now here we can write as y is equals to fx and this is equal to 1 divided with x.

01:52 Now we can write it as y is equals to 1 divided with x.

01:58 From here we can conclude that x is equal to 1 divided with y...

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