Check the definition below. Present another real-life situation to explain the meaning of functions.
The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation Ar². With each positive number r there is associated one value of A, and we say that A is a function of r.
Answer (1)
Define distance d as a function of time t with constant speed v .
Express the relationship as d = v \t .
Illustrate with an example: if v = 60 mph, then d = 60 t .
Conclude that distance traveled is a function of time.
Explanation
Understanding the Problem We want to illustrate the concept of a function with a real-life example, similar to how the area of a circle depends on its radius.
Defining the Relationship Consider the relationship between the distance traveled by a car and the time it travels, assuming the car moves at a constant speed.
Expressing the Relationship as a Function Let d be the distance traveled, t be the time traveled, and v be the constant speed of the car. The relationship between these variables is given by the equation: d = v \t Here, the distance d is a function of the time t , given a constant speed v .
Illustrating with an Example For example, if the car travels at a constant speed of 60 miles per hour, then the distance traveled is d = 60 t . This means that for every value of time t , there is a unique value of distance d . If t = 2 hours, then d = 60 \t 2 = 120 miles.
Concluding the Example In this scenario, the distance traveled is a function of time, where the function rule is defined by the constant speed.
Examples
Understanding functions is crucial in many real-world applications. For instance, consider a delivery service where the cost of delivery depends on the distance to the destination. The delivery cost can be modeled as a function of distance, helping the company optimize pricing and plan routes efficiently. Similarly, in manufacturing, the production cost can be modeled as a function of the number of units produced, aiding in cost management and production planning. Recognizing and utilizing functional relationships allows for better decision-making and optimization in various fields.
