Question

if the relationships below are given in the form (input, output), which pairing always describes a function?
○ (a persons age in years, that same persons height in inches)
○ (a persons weight in pounds, that same persons height in inches)
○ (a persons height in centimeters, that same persons height in inches)
○ (a persons telephone number, that same persons height in inches)

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input has exactly one output. So we need to check for each option if one input (first element of the pair) can have only one output (second element of the pair).

Step2: Analyze Option 1: (age, height)

A single age can correspond to different heights (for example, different people of the same age can have different heights, or even the same person at different times? Wait, no, the same person's age and height: but as a person grows, at a single age (input), do they have one height? Wait, actually, for a given person, at a specific age, there is one height, but the question is about "a person's age" – wait, no, the input is "a person's age in years" (so the input is an age value, not a specific person). Wait, no, the pairing is (input, output) where input is a person's age, output is that same person's height. Wait, maybe I misread. Let's re-express:

For option 1: Input is a person's age (so the age of a person), output is that same person's height. But a person has one age at a time, but the problem is about the relation: if we consider the input as age (a number), and output as height (a number), can one age have multiple heights? For example, different people of the same age can have different heights, but the pairing is "that same person's height" – wait, no, the input is "a person's age in years" (so the age of a particular person), output is "that same person's height". Wait, maybe the question is: for a given person, (age, height) – but the problem is phrased as "which pairing always describes a function" – so we need a relation where each input (first element) has exactly one output (second element), regardless of the person? Wait, no, the pairing is (input, output) where input is a person's [x], output is that same person's [y]. So for a function, for each input (x - value), there is exactly one output (y - value) for the same person? Wait, maybe the key is: for the relation, when the input is determined, the output is uniquely determined.

Let's check each option:

Option 1: (age, height). A person's age (input) – but a person can have different heights at the same age? No, at a specific age (like 10 years old), a person has one height (at that time). But wait, the problem is about the pairing: if we consider the input as age (a number), and output as height (a number), but different people with the same age can have different heights. Wait, no, the pairing is "that same person's height" – so the input is a person's age (so for a specific person, age is input, height is output). But the question is about the relation: does each input (age) map to exactly one output (height) for the same person? But the problem is asking "which pairing always describes a function" – so we need a relation where, for any input, there's exactly one output.

Option 2: (weight, height). A person's weight (input) and that same person's height (output). A person can have the same weight at different times with different heights (e.g., muscle gain vs fat loss), so one weight (input) can correspond to different heights (output) for the same person. So this is not a function.

Option 3: (height in cm, height in inches). The conversion from centimeters to inches is a linear conversion: \( \text{height in inches} = \frac{\text{height in centimeters}}{2.54} \). So for a given height in centimeters (input), there is exactly one height in inches (output), because the conversion is a fixed formula. So this is a function, since each input (cm height) has exactly one output (inch height) for the same person (or anyone, since the conversion is fixed).

Option 4: (telephone number, height). A telephone number (input) and that same person's height (output). A telephone number is unique to a person (usually), but the input is the telephone number, and the output is height. But the question is about the relation: does each telephone number (input) have exactly one height (output)? Well, a telephone number belongs to one person, so that person has one height, so maybe? Wait, no, the problem is: is the relation a function? Wait, no, the key is: for the input (telephone number), is there exactly one output (height)? Since a telephone number is associated with one person, that person has one height, so yes? But wait, no, the conversion in option 3 is a mathematical function, while option 4 is just a relation where each telephone number (input) has one height (output), but is that always true? Wait, but the question is which "always" describes a function. Let's re-examine.

Wait, the key is: in a function, each input must have exactly one output. Let's re-express each option:

Option 1: Input = age (of a person), Output = height (of that person). But a person's age (say, 20 years) – but different people of age 20 can have different heights, but the pairing is "that same person's height" – so for a specific person, age and height: but the input is "a person's age" (so the age of a person), output is "that same person's height". So for a given person, at a given age, there is one height. But the problem is about the relation as a function: if we consider the input as age (a number), and output as height (a number), but age can be the same for different people with different heights. Wait, maybe I misinterpret the pairing. The pairing is (input, output) where input is a person's [x], output is that same person's [y]. So it's a relation where the input is a property of a person, output is another property of the same person. So for the relation to be a function, for each input value (x - value of a person), there must be exactly one output value (y - value of that person).

So:

Option 1: x = age (of a person), y = height (of that person). Can a person have multiple heights at the same age? No, at a specific age (like 10 years old), a person has one height (at that time). But as a person grows, over time, the same age (wait, no, age increases over time). Wait, maybe the problem is considering the input as the age (a number) and the output as the height of the person at that age. But for a given age (input), different people can have different heights, but the pairing is "that same person's height" – so it's for a single person. Wait, this is confusing. Let's think of the function definition: a function is a set of ordered pairs where each first element is unique and maps to exactly one second element.

In option 3: height in cm (input) and height in inches (output). The conversion is a linear function, so for any height in cm (input), there is exactly one height in inches (output), because \( \text{inches} = \text{cm} / 2.54 \). So this is a mathematical function, so it always has one output per input.

In option 1: age (input) and height (output). A person's age (say, 15) – but a person at age 15 has one height, but different people at age 15 have different heights. But the pairing is "that same person's height" – so for a specific person, age and height: but the input is "a person's age" (so the age of a person), output is "that same person's height". So for a given person, each age (input) has one height (output), but the problem is about the relation as a function: if we consider the input as age (a number), and output as height (a number), but age can be the same for different people with different heights. Wait, no, the pairing is (input, output) where input is a person's age (so the age of a particular person), output is that same person's height. So for a particular person, age and height: but the function is defined as a relation where each input (age) has one output (height). But is that always true? For a person, at a specific age, there is one height, but over time, age increases, and height changes. Wait, maybe the problem is considering the input as the age (a value) and the output as the height of the person with that age. But this is ambiguous.

Option 3: height in cm to height in inches. Since the conversion is a fixed formula, each input (cm height) has exactly one output (inch height), because \( \text{inches} = \text{cm} / 2.54 \). So this is a function, because the output is uniquely determined by the input.

Option 4: telephone number to height. A telephone number is associated with one person, so each telephone number (input) has one height (output), but is that a function? Well, yes, but the key is "always" – the conversion in option 3 is a mathematical function, so it's always a function, while the others may not be.

Wait, let's re-express the function definition: a function is a relation where each input has exactly one output. So:

- Option 1: Input = age (of a person), Output = height (of that person). But age can be the same for different people with different heights (if we consider the input as age, not a specific person). Wait, no, the pairing is "a person's age" (so the age of a person, i.e., the age belonging to a person), so the input is the age of a person, output is the height of that person. So for a given person, age and height: but the problem is about the relation as a function. If we consider the domain as all possible (person, age) pairs, but no, the input is "a person's age in years" (so the age value, e.g., 25), and output is "that same person's height in inches" (so the height of the person with that age). Wait, this is confusing. Let's take an example.

Suppose we have a person A, age 20, height 60 inches. Person B, age 20, height 70 inches. So if the input is 20 (age), the output could be 60 (for A) or 70 (for B). So the input 20 has multiple outputs, so it's not a function.

Option 2: (weight, height). A person's weight (input) and that same person's height (output). A person can have the same weight at different times with different heights (e.g., muscle vs fat), so input (weight) can have multiple outputs (height), so not a function.

Option 3: (height in cm, height in inches). The conversion is \( \text{inches} = \frac{\text{cm}}{2.54} \). So for any height in cm (input), there is exactly one height in inches (output), because the formula is fixed. So this is a function, since each input has exactly one output.

Option 4: (telephone number, height). A telephone number (input) and that same person's height (output). A telephone number is unique to a person, so each telephone number (input) has one height (output), but is that a function? Well, yes, but the problem is "always" – the conversion in option 3 is a mathematical function, so it's always a function, while option 4 is a relation where each telephone number has one height, but is that necessarily always true? For example, if a person changes their height (unlikely, but possible), but the telephone number remains the same. Wait, no, the height is of "that same person", so if the person's height changes, the telephone number (input) would still have the current height (output). But the key is: in a function, the output must be uniquely determined by the input. In option 3, the output is uniquely determined by the input (cm height) via a fixed formula, so it's a function. In option 4, the output is determined by the person associated with the telephone number, but the input (telephone number) maps to one person, who has one height, so it's also a function? But that can't be, because the question is multiple choice, and only one is correct.

Wait, no, the key is: the pairing in option 3 is a mathematical function (conversion of units), so each input (cm height) has exactly one output (inch height), because the conversion is a fixed formula. In option 4, the telephone number is an input that maps to a person's height, but the relation is not a mathematical function, just a relation where each telephone number (input) has one height (output), but is that "always" a function? Wait, maybe the question is considering the input and output as properties of a person, so for the same person, the input and output. Let's re-express:

The pairing is (input, output) where input is a property of a person, output is another property of the same person. So we need that for the same person, each input value has exactly one output value.

Option 1: For the same person, age (input) and height (output). As a person ages, their height changes, but at a specific age (input), they have one height (output). Wait, but the input is "a person's age in years" – so for a specific person, at a specific age, there is one height. But the problem is about the relation as a function: if we consider the input as age (a number) and output as height (a number), but age can be the same for different people with different heights. But the pairing is "that same person's height", so it's for a specific person. So for a specific person, age and height: but the function is defined for a person, so each age (input) has one height (output). But this is not a function over all people, but the question is about "which pairing always describes a function" – maybe the intended answer is option 3, because the conversion is a mathematical function, so it's always a function, while the others may not be.

Yes, the key is that in option 3, the input (height in cm) and output (height in inches) have a fixed mathematical relationship (conversion factor), so each input has exactly one output. In the other options, the input does not uniquely determine the output (age can have different heights, weight can have different heights, telephone number is unique but the relation is not a mathematical function, but the conversion in option 3 is a function).

Answer:

C. (a person's height in centimeters, that same person's height in inches)