Question

A function is being used by the manager of a factory producing lawn mowers. The output of the function tells the manager how many lawn mowers will be made, using the number of workers for the input. What types of numbers are acceptable values for the input? A. Any number will do. B. Only positive numbers can be used. C. Only integers can be used. D. Only non-negative integers can be used.

Short answer

D. Only non-negative integers can be used.

Step-by-step solution

Understand the Context

The function determines the number of lawn mowers produced based on the number of workers. It's important to consider practical and logical constraints to identify acceptable input values.

Evaluate Possible Inputs

Identify whether any type of number (e.g., negative, positive, zero, fractions) can sensibly represent the number of workers in a factory setting.

Analyze Option A

Consider if any number will do. Negative numbers and fractions are not practical for representing a count of workers.

Analyze Option B

Assess if only positive numbers should be used. While we certainly need positive workers for production, this option excludes zero, which can represent a situation where no workers are present.

Analyze Option C

Review if only integers should be used. While fractions are not sensible (you can’t have half a worker), this option does not consider that negative numbers are also integers.

Analyze Option D

Examine if only non-negative integers can be used. Non-negative integers include zero and all positive whole numbers, which are practical and appropriate for counting workers.

Draw the Conclusion

Based on the analysis, only non-negative integers (0, 1, 2, …) make sense as they represent realistic counts of workers.

Key concepts

Non-negative Integers

When working with functions, especially in real-life scenarios such as a factory setting, understanding what values can serve as valid inputs is crucial. In our case with the lawn mower production function, non-negative integers provide the most logical set of inputs.

Non-negative integers are numbers that include zero and all positive whole numbers, such as 0, 1, 2, 3, and so on.
This set of numbers makes sense because you cannot have a negative number of workers, and fractions are not practical in counting people. Each worker must be considered as a whole entity. By restricting inputs to non-negative integers, we ensure that the input values are feasible and meaningful within the given context. This restriction avoids illogical scenarios, such as having -1 workers or 2.5 workers, which could otherwise lead to confusion or erroneous outputs.

Contextual Constraints

When applying a function to a real-world problem, contextual constraints help us determine the appropriate input values.
For example, the function here determines the number of lawn mowers produced based on the number of workers. Practical constraints guide us towards logical values for these inputs.
One crucial constraint is that the number of workers cannot be negative because, in reality, you cannot have a negative quantity of people. Similarly, it does not make sense to have fractional values, such as 2.5 workers.
Contextual constraints ensure that the function behaves logically and produces meaningful outputs. By taking into account real-life scenarios and constraints, we can narrow down the acceptable range of input values to those that make sense. This facilitates the creation of accurate and reliable functions that reflect real-world conditions efficiently.

Function Output

The output of a function is determined by the input values we provide. In the context of our factory example, the function's output indicates the number of lawn mowers produced based on the number of workers.
Ensuring that our input values are valid (non-negative integers) directly impacts the accuracy and reliability of the function's output. If we input a number like -3 or 2.5 into our function, the output would be nonsensical since these values do not represent real-world quantities of workers.
Therefore, the output will only be meaningful if the inputs are constrained appropriately.
By applying non-negative integers as input values, we guarantee that the function produces realistic and applicable results. This way, the manager can correctly understand how many lawn mowers will be made, aiding in logical and effective decision-making processes in the production environment.