By "Grace Williams", Updated Aug 30, 2022

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In mathematics, a function maps each input value x to a single output value y. While each x has only one corresponding y, a given y may be produced by multiple x values. The set of all admissible x values is called the domain, and the set of all attainable y values is the range. Theoretical domains and ranges consider every mathematically possible solution, whereas practical domains and ranges restrict the values to realistic bounds relevant to a specific context.

Step 1 – Build the Function

Translate a word problem into a function that inherently defines the practical domain and range. For instance, consider Anna, who babysits for the Smith family. She earns a flat $10 for arriving, plus $2 per hour of care, with a maximum of 10 hours. Defining the total earnings as y and the unknown hours worked as x, the function becomes:

y = 10 + 2x

Step 2 – Identify the Domain

The domain reflects all feasible values of x. Anna may work from 0 up to 10 hours, so the domain is expressed as the inequality:

0 ≤ x ≤ 10

Step 3 – Determine the Range

Substitute the extreme domain values into the function to compute the corresponding earnings. For x = 0:

y = 10 + 2(0) = 10

For x = 10:

y = 10 + 2(10) = 30

Thus, the practical range is:

10 ≤ y ≤ 30