Each person who enters a store receives a $5 off coupon. In the equation y = 5x, y is the total value of the coupons given out by the store, and x is the number of people receiving the coupons.
Wen says the function is continuous, because the number of people is unlimited. Is Wen right or wrong? Explain.
Answer (2)
Define continuous and discrete functions.
Determine that the number of people ( x ) must be a non-negative integer.
Conclude that the function y = 5 x is discrete because x can only take on discrete values.
State that Wen is wrong because the function is discrete, not continuous, as continuity requires the variable to take on any value within a range, including fractional values. Wen is wrong.
Explanation
Problem Analysis Let's analyze the problem. We are given the equation y = 5 x , where y represents the total value of coupons given out and x represents the number of people receiving coupons. Wen claims that this function is continuous because the number of people is unlimited. We need to determine if Wen is correct.
Continuous vs. Discrete Functions A continuous function is a function whose graph can be drawn without lifting your pen from the paper. In other words, a continuous function can take on any value within a given range, including fractional values. A discrete function , on the other hand, is a function whose graph consists of isolated points. Discrete functions typically deal with integer values only.
Possible Values of x In this scenario, x represents the number of people receiving coupons. Can we have a fractional number of people? No, we cannot. The number of people must be a non-negative integer (0, 1, 2, 3, ...). Therefore, x can only take on discrete values.
Function Type Since x can only take on discrete values, the function y = 5 x is a discrete function, not a continuous function. The graph of this function would be a series of isolated points, where each point represents a possible number of people and the corresponding total value of coupons.
Wen's Reasoning Wen's reasoning is incorrect. While the number of people might be unlimited in theory (you could have any number of people enter the store), the number of people must still be an integer. The fact that the number of people is 'unlimited' does not make the function continuous. Continuity requires that the variable can take on any value within a range, including fractional values, which is not the case here. Therefore, Wen is wrong.
Examples
Imagine you're organizing a school play and selling tickets for $5 each. The total revenue you collect depends on the number of tickets sold. Since you can only sell whole tickets (you can't sell half a ticket), the relationship between the number of tickets sold and the total revenue is discrete. This is similar to the coupon problem, where you can only give coupons to whole people, making the function discrete, not continuous. Understanding the difference between continuous and discrete functions helps in modeling real-world scenarios accurately.
Wen is incorrect; the function y = 5 x is discrete, not continuous. This is because the variable x , representing the number of people, must be a non-negative integer. Discrete functions consist of isolated points, which applies to this scenario since you can't have a fraction of a person receiving a coupon.
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