Relating Domain to Graph and Context

Deck 1 of 2: Natural and Contextual Domain

In this deck:

• Find domain restrictions from algebraic rules
• Narrow domain further using real-world context

What You Will Learn Today

1. Find the natural domain from algebraic rules
2. Find the contextual domain from real-world constraints
3. Read domain from a graph using vertical scan
4. Distinguish discrete from continuous domains
5. Reconcile all three domain perspectives
6. Write domains in interval, set-builder, and roster notation

Can ANY x Work Here?

Which of these functions accepts every real number as an input?

Think before the next slide — which ones have restrictions?

Natural Domain — Two Restriction Types

The natural domain is every real number where the formula produces a defined output.

Two things that can cause problems:

1. Denominators — cannot equal zero
2. Even roots — radicand must be non-negative

Everything else (linear, polynomial, odd roots) accepts all reals.

Setting the Denominator Not Equal to Zero

Step 1: Identify the denominator:

Step 2: Set it not equal to zero:

Step 3: Solve:

Domain: all real numbers except

Setting the Radicand Greater Than Zero

Step 1: Identify the radicand:

Step 2: Set it ≥ 0:

Step 3: Solve: , so

Domain:

Applying Both Restrictions at the Same Time

Need BOTH conditions to hold simultaneously:

• Radicand:
• Denominator:

Domain:

Your Domain Checklist for Every Function

Apply both checks — a function may have both issues.

Check: Find the Natural Domain Here

Find the natural domain of:

Write your answer in interval notation before the next slide.

Answer: Excluding Two Values from the Domain

Set denominator ≠ 0:

Domain:

What the Math Allows vs. What Makes Sense

The natural domain tells us what the math allows.

But real-world functions also have context that limits inputs further.

The contextual domain is always a subset of the natural domain.

Context restricts. Context never expands.

Contextual Domain — Three Common Restrictions

When a function models a real situation, ask: what does the variable represent?

1. Non-negativity — time, length, count must be ≥ 0
2. Integrality — countable things must be whole numbers
3. Upper/lower bounds — physical limits cap the domain

Engine Assembly: The Standard's Example

Natural domain: all real numbers (algebra allows it)
Contextual domain: — positive integers only

Candle Function: Continuous with Upper Bound

Natural domain: all reals

Contextual restrictions:

• Time must be non-negative:
• Candle exists until it burns out:

Contextual domain: — continuous interval

Bus Fare Limits Passengers to Whole Numbers

Natural domain: all reals

Contextual restrictions:

• must be a whole number (can't have half a passenger)
• (bus capacity)

Contextual domain: — discrete set

Ball Height Defines Its Own Upper Bound

Contextual restrictions:

• (before the throw, is meaningless)
• (ball stays above ground)

Solve : seconds

Contextual domain: — continuous

Check: State the Water Taxi Domain

A water taxi charges $8 per passenger and holds at most 12 passengers.

State:

1. The natural domain
2. The contextual domain
3. Is the domain discrete or continuous?

Write your answers before the next slide.

Water Taxi Domain Answer Revealed

1. Natural domain: all real numbers (formula works for any )
2. Contextual domain:
   • Passengers are whole numbers
   • Maximum of 12 (boat capacity)
3. Discrete — individual dots when graphed; 13 points total

Summary: Three Contextual Restriction Types

Context always restricts the natural domain — never expands it.

Key Takeaways from Deck One

✓ Natural domain: denominators ≠ 0, radicands ≥ 0

✓ Contextual domain: a subset — context always restricts further

✓ Context wins in modeling: ask what the variable represents

Not every function accepts all reals — run the checklist

Domain = valid inputs, not acceptable outputs

Coming Up in Deck Two

Domain from Graphs and Discrete vs. Continuous

• Reading domain from a graph — the vertical scan technique
• Solid dots, open circles, asymptotes, and holes
• Continuous functions vs. discrete functions
• Synthesis: three perspectives on one function

Applies LOs 3, 4, 5, and 6