/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 \mathrm{A}=\\{-1,0,1,2\\}, \math... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 29

\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

Short Answer

Expert verified
However, our initial analysis was incorrect, as the function f is neither one-one nor onto. The correct answer is: Function f is not a one-one, onto, or bijective function.

Step by step solution

01

(1) Define one-one function.

A one-one function (also known as an injective function) is a function where each element in the domain (set A) maps to a unique element in the codomain (set B). In other words, if f(x) = f(y), then x=y.
02

(2) Test f for being one-one.

Using the definition of f, f(x) = x^2. We must check if there exists x, y in set A, such that f(x)=f(y) and x鈮爕. Let x = -1 and y = 1, then f(x) = (-1)^2 = 1 and f(y) = (1)^2 = 1. Since f(-1) = f(1) but -1鈮�1, the function f is not one-one.
03

(3) Define onto function.

An onto function (also known as a surjective function) is a function where for every element y in the codomain (set B), there exists an element x in the domain (set A) such that f(x)=y.
04

(4) Test f for being onto.

Using the definition of f, for each element in set B, we must determine if there exists an element in set A that maps to it. 1. For y=0, x=0, f(x) = (0)^2 = 0 2. For y=1, x=1 or x=-1, f(x) = (1)^2 = 1 or f(x) = (-1)^2 = 1 3. For y=2, there is no x in set A such that f(x) = x^2 = 2 Since there is no element in the domain (set A) that maps to the element y=2 in the codomain (set B), the function f is not onto.
05

(5) Define bijective function.

A bijective function is a function that is both one-one (injective) and onto (surjective).
06

(6) Test f for being bijective.

Since we have already determined that function f is neither one-one nor onto, it cannot be a bijective function.
07

(7) Define function.

A function is a relation between two sets A and B such that each element of set A has a unique image in set B. In our case, the function is well-defined, as each element in set A maps to a unique element in set B.
08

(8) Conclusion.

Based on our analysis and testing, the correct answer is: Function f is not one-one or onto, therefore the answer is (1) only one-one function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

One-One Function
A one-one function, also known as an injective function, is a fundamental concept in mathematics. In such a function, every element in the domain has a unique mapping to the codomain.
This means no two different elements in the domain map to the same element in the codomain. This ensures that if \( f(x) = f(y) \), then \( x = y \).
  • For example, imagine two elements, \( x = 2 \) and \( y = 3 \) in set A. If both map to 4 in set B, then the function is not injective.
  • However, if \( x = 2 \) maps to 4 and \( x = 3 \) maps to 5, then it is one-one.
Testing whether a function is one-one involves verifying this uniqueness criteria across the domain.
Onto Function
An onto function, or surjective function, is characterized by its thorough coverage of the codomain.
Essentially, every element in the codomain has at least one pre-image in the domain.
This ensures comprehensive mapping.
  • Consider a codomain that has elements 1, 2, and 3.
  • An onto function from a domain will map at least one element from the domain to each of these codomain values.
This means we need to verify for each codomain element if there is a corresponding domain element that maps to it through the function. If any codomain element is missing a pre-image, the function is not onto.
Bijective Function
A bijective function is a perfect marriage between the one-one and onto functions.
It is both injective and surjective, meaning every element in the domain maps uniquely and comprehensively to elements in the codomain.
  • Every domain element maps to a unique codomain element (like one-one).
  • Every codomain element is mapped to by some domain element (like onto).
Testing for bijectivity means confirming that both one-one and onto conditions are satisfied.
So, a bijection ensures each element in both sets has a unique counterpart in the other set, establishing a perfect correlation.
Function Definition
The concept of a function is pivotal in mathematics, representing a relationship between two sets, typically called the domain and codomain.
A function assigns exactly one output (in the codomain) for each input (in the domain).
  • Imagine a vending machine, where pressing a button (domain) gives you a particular snack (codomain).
  • Each action should provide one result, ensuring clarity and reliability in the output.
This definition implies a clear and consistent relationship: no input should produce more than one result.
Understanding this basic rule is essential when analyzing or constructing functions.

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