Relation 3
$\lbrace(-4,7),(-6,-4),(7,7),(-5,-4)\rbrace$
Function
Not a function
Answer (1)
Identify the domain and range of the relation.
Check if each element in the domain maps to a unique element in the range.
If each element in the domain maps to a unique element in the range, then the relation is a function.
The given relation is a function. F u n c t i o n
Explanation
Understanding the Problem We are given a relation represented as a set of ordered pairs and we need to determine whether this relation is a function or not. A relation is a function if each element in the domain (the set of first elements in the ordered pairs) maps to a unique element in the range (the set of second elements in the ordered pairs). In other words, for every x-value, there can only be one y-value.
Identifying Domain and Range Let's identify the domain and range of the given relation: ${(-4,7),(-6,-4),(7,7),(-5,-4)} . T h e d o maini s t h ese t o f a ll f i rs t e l e m e n t s : {-4, -6, 7, -5} . T h er an g e i s t h ese t o f a ll seco n d e l e m e n t s : {7, -4}$.
Checking for Uniqueness Now, let's check if any element in the domain maps to more than one element in the range. - The element -4 maps to 7. - The element -6 maps to -4. - The element 7 maps to 7. - The element -5 maps to -4. Each element in the domain maps to only one element in the range. Therefore, the relation is a function.
Conclusion Since each element in the domain maps to a unique element in the range, the given relation is a function.
Examples
In real life, functions are used to model relationships between different quantities. For example, the relationship between the number of hours worked and the amount of money earned can be represented as a function. If each hour worked corresponds to a unique amount of money earned, then the relationship is a function. Similarly, the relationship between the temperature of an oven and the time it takes to bake a cake can be modeled as a function. Understanding functions helps us predict and analyze these relationships.
