Math Problem Statement

Determine whether the relation defines a​ function, and give the domain and range. ​{(9​, 3​), ​(8​, 7​), ​(5​, 1​), ​(2​, 4​)}

Solution

To determine whether the given relation defines a function, we need to verify if each input (or first element) is associated with exactly one output (or second element). In other words, for every distinct xx-value, there should be a unique yy-value.

The relation is:
{(9,3),(8,7),(5,1),(2,4)}\{(9, 3), (8, 7), (5, 1), (2, 4)\}

Step 1: Check if the relation is a function.

  • In the ordered pairs, the first elements are 9,8,5,29, 8, 5, 2, which are all distinct.
  • Since each xx-value has exactly one corresponding yy-value, the relation does define a function.

Step 2: Determine the domain.

The domain is the set of all possible xx-values (first elements) in the relation: Domain={9,8,5,2}\text{Domain} = \{9, 8, 5, 2\}

Step 3: Determine the range.

The range is the set of all possible yy-values (second elements) in the relation: Range={3,7,1,4}\text{Range} = \{3, 7, 1, 4\}

Conclusion:

  • The relation defines a function.
  • Domain: {9,8,5,2}\{9, 8, 5, 2\}
  • Range: {3,7,1,4}\{3, 7, 1, 4\}

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:

  1. What conditions must a relation meet to be considered a function?
  2. How do we determine the domain and range for more complex relations?
  3. Can a relation still be a function if two different yy-values map to the same xx-value?
  4. What are the differences between injective, surjective, and bijective functions?
  5. How would the answer change if one xx-value was associated with two different yy-values?

Tip: Always check the uniqueness of each xx-value's corresponding yy-value to confirm if a relation is a function.

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Math Problem Analysis

Mathematical Concepts

Functions
Relations
Domain and Range

Formulas

-

Theorems

Definition of a Function

Suitable Grade Level

Grades 8-10