Problem 132
Question
Consider a rule \(f(x)=2 x-3\). Whether \(f: N \rightarrow N\) is a function or not?
Step-by-Step Solution
Verified Answer
The given function \(f(x) = 2x - 3\) does not map natural numbers to natural numbers, as seen when x = 1, it outputs -1 which is not a natural number. Therefore, \(f: \mathbb{N} \rightarrow \mathbb{N}\) is not a valid function.
1Step 1: Recall definition of function, domain and range
A function is a rule that assigns each element of its domain to exactly one element of its range. The domain is the set of input values (in this case, natural numbers) and the range is the set of output values (also natural numbers).
For a given function to be considered as \(f: \mathbb{N} \rightarrow \mathbb{N}\), we must ensure that when a natural number is inputted as 'x', the function should output a natural number as 'f(x)' as well.
2Step 2: Analyze the function
The given function is \(f(x) = 2x - 3\). Let's see if it maps natural numbers to natural numbers.
When x = 1 (smallest natural number), the function gives:
\(f(1) = 2(1) - 3 = 2 - 3 = -1\)
Since the output is not a natural number but rather an integer, this function does not map natural numbers to natural numbers.
3Step 3: Conclusion
Based on the analysis, the given function \(f(x) = 2x - 3\) does not map natural numbers to natural numbers. Therefore, \(f: \mathbb{N} \rightarrow \mathbb{N}\) is not a valid function.
Key Concepts
Definition of FunctionNatural Numbers MappingAnalysis of Function
Definition of Function
Understanding the definition of a function is foundational in mathematics. A function is much like a vending machine that dispenses exactly one item for each coin inserted. Mathematically speaking, a function is a rule that associates each element in a set, called the domain, to exactly one element in another set, often termed the range.
This strict one-to-one relationship means that for every input there is a single, well-defined output. When we denote a function as \(f: \mathbb{A} \rightarrow \mathbb{B}\), the \(\mathbb{A}\) represents the domain and \(\mathbb{B}\) represents the range. The critical aspect of a function is that each input from the domain \(\mathbb{A}\) is connected to exactly one output in the range \(\mathbb{B}\), with no input paired with multiple outputs.
This strict one-to-one relationship means that for every input there is a single, well-defined output. When we denote a function as \(f: \mathbb{A} \rightarrow \mathbb{B}\), the \(\mathbb{A}\) represents the domain and \(\mathbb{B}\) represents the range. The critical aspect of a function is that each input from the domain \(\mathbb{A}\) is connected to exactly one output in the range \(\mathbb{B}\), with no input paired with multiple outputs.
Natural Numbers Mapping
Mapping in mathematics refers to the process of associating elements from one set to elements in another set. Specifically, 'natural numbers mapping' involves taking elements from the set of natural numbers (denoted as \(\mathbb{N}\)), such as 1, 2, 3, and so on, and finding their partners in a second set under the rules of a function.
It’s important to stress that natural numbers are the positive integers, starting from 1 and moving upwards indefinitely. When talking about functions, they often take on the role of the domain. The function described above, \(f(x) = 2x - 3\), is under analysis to determine if it properly maps natural numbers to natural numbers. This means for every natural number \(x\), \(f(x)\) should also be a natural number. If any natural number \(x\) produces an output outside of the natural numbers, the rule does not qualify as a \(f: \mathbb{N} \rightarrow \mathbb{N}\) function.
It’s important to stress that natural numbers are the positive integers, starting from 1 and moving upwards indefinitely. When talking about functions, they often take on the role of the domain. The function described above, \(f(x) = 2x - 3\), is under analysis to determine if it properly maps natural numbers to natural numbers. This means for every natural number \(x\), \(f(x)\) should also be a natural number. If any natural number \(x\) produces an output outside of the natural numbers, the rule does not qualify as a \(f: \mathbb{N} \rightarrow \mathbb{N}\) function.
Analysis of Function
Breaking down a function’s behavior and characteristics is referred to as the 'analysis of function'. This process involves checking the function against various criteria to ensure it satisfies the conditions for being a true function.
When we analyze, we're interrogating the function at fundamental levels - such as whether the output for every input in the domain is valid for the range. Considering the exercise where \(f(x) = 2x - 3\), analysis reveals a mismatch for natural numbers, as inserting 1 yields -1, which is not in the set of natural numbers.
Such an examination extends to other aspects such as continuity, rate of growth, and more complex behavior which could be subject to further mathematical exploration. Function analysis is a critical skill in mathematics as it enables one to predict and understand how a function behaves and reacts to different inputs.
When we analyze, we're interrogating the function at fundamental levels - such as whether the output for every input in the domain is valid for the range. Considering the exercise where \(f(x) = 2x - 3\), analysis reveals a mismatch for natural numbers, as inserting 1 yields -1, which is not in the set of natural numbers.
Such an examination extends to other aspects such as continuity, rate of growth, and more complex behavior which could be subject to further mathematical exploration. Function analysis is a critical skill in mathematics as it enables one to predict and understand how a function behaves and reacts to different inputs.
Other exercises in this chapter
Problem 130
Let \(A=\\{1,2,3,\\}, B=\\{2,3,4\),\(} then which of the following is function from A\) to \(B:-\) i. \(f_{1}=\\{(1,2),(1,3),(2,3),(3,3)\\} .\) ii. \(f_{2}=\\{(
View solution Problem 131
Let \(A=\\{-2,-1,0,1,2\\}\) and \(B=\\{0,1,2,3,4,5,6\\}\) and a rule \(f(x)=x^{2}\). Whether \(f: A \rightarrow B\) is a function or not? If yes, find range of
View solution Problem 133
Let \(A=\\{-2,-1,0,1,2\\}\) and \(f: A \rightarrow I\) given by \(f(x)=x^{2}-2 x-3\). Find range of \(f\). Also find preimages of \(6,-3\) and 5. \\{Ans. \(f(A)
View solution Problem 134
Given \(A=\\{-1,0,2,5,6,11\\}\) and \(B=\\{-2,-1,0,18,28,108\\}\) and \(f(x)=x^{2}-x-2\). Find \(f(A)\). Whether \(f(A)=B\) or not.
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