Understanding Function Domains and Ranges in Real-World Applications
The concept of domain and range in mathematical functions plays a crucial role in analyzing real-world scenarios. Let's explore various examples to deepen our understanding of these fundamental concepts.
Definition: Domain represents all possible input values (x-values) of a function, while range includes all possible output values (y-values).
When examining a function through its mapping diagram, determining the domain becomes straightforward. For instance, in a discrete function where specific points are mapped, the domain consists of all x-values that have corresponding y-values. Consider a function with points at x = -4, -1, 0, 2, and 7. These specific x-coordinates form the complete domain of the function.
Real-world applications provide excellent contexts for understanding function relationships. Take a banquet hall rental scenario: The total cost depends on the rental duration, with a rate of 85perhalfhourplusa50 cleaning fee, capped at 4 hours. This creates a linear function where the domain is limited to realistic rental durations (0 to 4 hours), and the range extends from the minimum cost (50)tothemaximumpossiblecharge(730).
Example: In the banquet hall scenario:
- Minimum cost = $50 (cleaning fee only)
- Maximum cost = (4 hours × 170/hour)+50 = $730