A={-1,0,1,2}, B={0,1,2} and f: A →B defined by f(x)=x^2, then f is (1) only one-one function (2)

step 1 Hello students in this question set A and set B two sets are given and we have a function F is d

A={-1,0,1,2}, B={0,1,2} and f: A →B defined by f(x)=x^2,...

\mathrm{A}=\{-1,0,1,2\}, \mathrm{B}=\{0,1,2\}$ and
$\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$, then $\mathrm{f}$ is
(1) only one-one function
(2) only onto function
(3) bijective
(4) not a function

Key Concepts

Surjective (Onto) Function

A surjective function, or onto function, ensures that every element in the codomain is the image of at least one element from the domain. This concept is essential in determining whether the function covers the entire codomain, which is a critical part of understanding the overall coverage of the function's mapping.

Injective (One-to-One) Function

An injective function, or one-to-one function, is one where distinct elements in the domain map to distinct elements in the codomain. This property ensures that no two different inputs produce the same output, making it important when examining the uniqueness of mappings.

Bijective Function

A bijective function is one that is both injective and surjective. This means every element in the domain maps to a unique element in the codomain and every element of the codomain is covered, establishing a perfect one-to-one correspondence. Bijective functions are central in many mathematical areas because they guarantee the existence of inverses.

Function

A function is a mapping between two sets where every element in the domain is assigned exactly one element in the codomain. This concept ensures that for each input there is a unique output, and it is fundamental in analyzing the behavior and properties of mathematical relations.

Domain and Codomain

The domain of a function is the set of all possible inputs, while the codomain is the set of potential outputs as defined by the function. Understanding both is crucial because a function must assign each element from the domain to an element in the codomain in order to be well-defined.

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