Question
If the relationships below are given in the form (input, output), which pairing always describes a function?
(a person’s age in years, that same person’s height in inches)
(a person’s weight in pounds, that same person’s height in inches)
(a person’s height in centimeters, that same person’s height in inches)
(a person’s telephone number, that same person’s height in inches)
(a person’s age in years, that same person’s height in inches)
(a person’s weight in pounds, that same person’s height in inches)
(a person’s height in centimeters, that same person’s height in inches)
(a person’s telephone number, that same person’s height in inches)
Answers
GPT-4o mini
To determine which pairing describes a function, we need to ensure that for every input, there is exactly one output. Let's analyze each pairing:
1. **(a person’s age in years, that same person’s height in inches)**: The height of a person can change with age, and different people can have the same age but different heights. This does not guarantee that a single input (age) produces one output (height).
2. **(a person’s weight in pounds, that same person’s height in inches)**: Similar to age, different people can have the same weight but different heights. Therefore, this does not guarantee that there is one output for each input.
3. **(a person’s height in centimeters, that same person’s height in inches)**: The relationship between height in centimeters and height in inches is a direct conversion (1 inch = 2.54 centimeters). For every height in centimeters, there is exactly one corresponding height in inches, making this a function.
4. **(a person’s telephone number, that same person’s height in inches)**: A person can have a unique telephone number, but they can have different heights. This does not guarantee that each telephone number corresponds to one unique height, hence this does not describe a function.
From this analysis, **(a person’s height in centimeters, that same person’s height in inches)** always describes a function.
1. **(a person’s age in years, that same person’s height in inches)**: The height of a person can change with age, and different people can have the same age but different heights. This does not guarantee that a single input (age) produces one output (height).
2. **(a person’s weight in pounds, that same person’s height in inches)**: Similar to age, different people can have the same weight but different heights. Therefore, this does not guarantee that there is one output for each input.
3. **(a person’s height in centimeters, that same person’s height in inches)**: The relationship between height in centimeters and height in inches is a direct conversion (1 inch = 2.54 centimeters). For every height in centimeters, there is exactly one corresponding height in inches, making this a function.
4. **(a person’s telephone number, that same person’s height in inches)**: A person can have a unique telephone number, but they can have different heights. This does not guarantee that each telephone number corresponds to one unique height, hence this does not describe a function.
From this analysis, **(a person’s height in centimeters, that same person’s height in inches)** always describes a function.