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Interpret domain and range in context

Domain is the set of permissible input values for a function in a given situation, while range is the set of corresponding output values; both are interpreted in context, including restrictions such as nonnegative time, whole-number quantities, meaningful measurement intervals, and endpoint inclusion. The relationship is identified from equations, tables, graphs, or verbal descriptions, with attention to units and the distinction between algebraically possible values and values relevant to the situation; abstract set-theoretic, complex-valued, and other advanced domain–range treatments are not included.

Detailed Explanation: Interpret domain and range in context

A function’s domain is the set of input values that make sense in a situation. Its range is the set of output values produced by those inputs.

When interpreting domain and range, ask:

  • What does the input represent?
  • What restrictions does the situation give?
  • Are the inputs continuous measurements or whole-number quantities?
  • What outputs result from the allowed inputs?
  • Should endpoints be included?

Example

A school sells tickets for a concert. Each ticket costs $8, and the school must pay $500 in fixed costs. The concert hall holds 200 people.

Let xx be the number of tickets sold. The school’s profit is

P(x)=8x−500.P(x)=8x-500.

Find the domain and range in context.

Step 1: Interpret the input

The input xx represents the number of tickets sold.

Tickets are sold as whole tickets, so xx cannot be 2.52.5 or any other decimal. The school can sell at least 00 tickets and at most 200200 tickets because of the hall’s capacity.

Therefore, the domain is

{0,1,2,…,200}\{0,1,2,\ldots,200\}

tickets.

The endpoints are included because selling 00 tickets and selling 200200 tickets are both possible.

Step 2: Find the output values

The output P(x)P(x) represents profit in dollars.

For the smallest input, x=0x=0:

P(0)=8(0)−500=−500.P(0)=8(0)-500=-500.

The school loses $500 if it sells no tickets.

For the largest input, x=200x=200:

P(200)=8(200)−500=1100.P(200)=8(200)-500=1100.

The school makes $1,100 if it sells all 200 tickets.

Because xx must be a whole number, the profit increases by $8 for each additional ticket. Thus, the range is

{−500,−492,−484,…,1092,1100}\{-500,-492,-484,\ldots,1092,1100\}

dollars.

Equivalently, the range can be written as

{8x−500∣x∈{0,1,2,…,200}}.\{8x-500\mid x\in\{0,1,2,\ldots,200\}\}.

So, in context:

  • Domain: {0,1,2,…,200}\{0,1,2,\ldots,200\} tickets
  • Range: {−500,−492,−484,…,1100}\{-500,-492,-484,\ldots,1100\} dollars

Although the equation could be evaluated for any real number xx, decimal numbers of tickets do not make sense in this situation. Therefore, the context—not just the equation—determines the domain and range.

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