Question

if the relationships below are given in the form (input, output), which pairing always describes a function?

a (a persons age in years, that same persons height in inches)
b (a persons weight in pounds, that same persons height in inches)
c (a persons height in centimeters, that same persons height in inches)
d (a persons telephone number, that same persons height in inches)

Explanation:

⚡ Using what you learned: definition of a function

Step 1: Apply the definition of a function

A relation is a function if each unique input maps to exactly one output.

Step 2: Evaluate the options

• Option A: Input is a person's age in years. Multiple people can have the same age (e.g., 25 years old) but different heights. This means one input maps to multiple different outputs. Not a function.
• Option B: Input is a person's weight in pounds. Multiple people can weigh the same amount (e.g., 150 lbs) but have different heights. Not a function.
• Option C: Input is a person's height in centimeters. Since centimeters and inches both measure the exact same physical length, a specific height in centimeters converts to exactly one unique height in inches:

$$ \text{height in inches} = \frac{\text{height in centimeters}}{2.54} $$

Each unique input has exactly one output. This is always a function.

• Option D: Input is a person's telephone number. Multiple people can share a telephone number (e.g., a landline shared by family members), and those family members can have different heights. Not a function.

Answer:

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