Verified Answer :

  • Question 9: The relation is a function because each element in the domain \(\{1, 3, 4, 5, 7\}\) maps to exactly one element in the range.
  • Questions 10 & 11: These cannot be solved as they lack the necessary input data.
  • Final Answer:
  • Question 9: \(\boxed{\text{Yes}}\)
  • Question 10: \(\boxed{\text{Incomplete}}\)
  • Question 11: \(\boxed{\text{Incomplete}}\)

Explanation

  1. Analyzing Question 9 for Function Properties

The relation is defined as the set of ordered pairs \(S = \{(1,0), (3,7), (4,4), (5,-2), (7,0)\}\). A relation is a function if and only if each element in the domain is associated with exactly one unique element in the range. This means that no two ordered pairs can have the same first coordinate (x-value) with different second coordinates (y-values). We examine the set \(S\):

  • For \(x = 1\), the output is \(y = 0\).
  • For \(x = 3\), the output is \(y = 7\).
  • For \(x = 4\), the output is \(y = 4\).
  • For \(x = 5\), the output is \(y = -2\).
  • For \(x = 7\), the output is \(y = 0\).

Each x-value appears exactly once in the set of ordered pairs. Since every input (domain element) maps to exactly one output (range element), the relation is a function. Note that while multiple inputs may map to the same output (e.g., \(1\) and \(7\) both map to \(0\)), this does not violate the definition of a function.

  1. Addressing Incomplete Questions 10 and 11

Questions 10 and 11 do not provide the necessary relations, graphs, or data required to perform an analysis. As such, it is impossible to determine whether they represent functions or to define their respective domains and ranges. Without the specific sets, equations, or visual representations, these problems remain incomplete.

Examples

In real life, identifying functions is essential in data processing and computer science. For example, consider a database where each unique User ID (domain) must be associated with exactly one email address (range). If a User ID were linked to multiple email addresses, the system would face errors in authentication or communication. By applying the definition of a function—where every input has exactly one output—you can ensure that your data structures and logic remain consistent and reliable, much like checking the ordered pairs in Question 9.

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