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Relating Domain
to Context & Graphs

Learn how to identify and interpret the domain of a function from its graph, equation, and the real-world context it models.

Algebra 2

What Is the Domain?

The domain of a function is the complete set of input values (usually \( x \)) for which the function is defined. It answers the question: "What values of \( x \) can I plug in?"

Mathematical Domain
All \( x \)-values where the formula produces a valid output: no division by zero, no negative square roots, etc.
Contextual Domain
The subset of the mathematical domain that makes sense in a real-world scenario: no negative lengths, whole engines only, etc.

Reading Domain from a Graph

On a graph, the domain is the horizontal extent: the set of all \( x \)-values covered by the curve. Look left to right and identify where the graph starts and stops.

x y 1 2 3 4 5 6 7
Domain: \([1, 7]\)

Why Context Changes the Domain

A formula may accept any real number mathematically, but the real-world situation often restricts the inputs. Context can impose restrictions like non-negative values, whole numbers, or upper bounds.

The Domain Rule

To determine the domain, consider both the mathematical restrictions and the real-world constraints, then take the intersection.

Mathematical Domain Contextual Constraints Actual Domain
\( \text{No } \div 0 \)
\( \text{No } \sqrt{\text{neg}} \)
\( t \ge 0 \)
\( n \in \mathbb{Z} \)

Types of Domain Restrictions

Division by Zero
Exclude any \( x \) that makes a denominator equal to zero.
Even Roots
The radicand under \( \sqrt{\phantom{x}} \) must be \( \geq 0 \).
Logarithms
The argument of \( \ln(x) \) must be \( > 0 \).
Context Bounds
Physical limits such as time \( \geq 0 \), count \( \in \) whole numbers, lengths \( > 0 \).

Mathematical vs Contextual Domain

Mathematical Domain Contextual Domain
Based on Algebraic rules only Real-world meaning
Example for \( f(x)=\sqrt{x} \) \( x \geq 0 \) Maybe \( 0 \leq x \leq 100 \)
Includes All valid algebra inputs Only inputs that make physical sense
Notation Intervals, set-builder Often discrete or bounded

How to Determine the Domain

1
Start with all reals. Assume \( x \in (-\infty, \infty) \) unless something says otherwise.
2
Apply math restrictions. Remove values causing division by zero, negative radicands, or non-positive log arguments.
3
Apply context restrictions. Narrow further based on physical reality: non-negative time, whole-number counts, maximum capacity, etc.
4
Express the domain. Use interval notation, set-builder notation, or a written description as appropriate.

Example 1: Ball Toss

A ball is thrown upward. Its height is modeled by \( h(t) = -16t^2 + 48t + 5 \), where \( t \) is time in seconds. What is the domain?

t (sec) h 0 1 2 3 10 20 30 40
\( h=0 \)
 peak
Domain: \([0, \approx 3.1]\)

Example 2: Factory Output

A factory function \( C(n) = 50n + 200 \) gives the cost to produce \( n \) widgets. The factory can make at most 500 widgets per day. What is the domain?

n C 0 100 200 300 400 500
Domain: \(\{0, 1, 2, ..., 500\}\)

Example 3: Square Root Function

Find the domain of \( f(x) = \sqrt{2x - 6} \) and identify it on the graph.

x f(x) 0 5 10 15 undefined
\( x=3 \)
Domain: \([3, \infty)\)

Example 4: Rational Function

Find the domain of \( g(x) = \dfrac{1}{x - 2} \).

x g(x) -2 -1 0 1 2 3 4 5
\( x=2 \)

Interval Notation Guide

Notation Meaning Endpoint Included?
\( [a, b] \) \( a \leq x \leq b \) Both included
\( (a, b) \) \( a < x < b \) Neither included
\( [a, b) \) \( a \leq x < b \) Left only
\( [a, \infty) \) \( x \geq a \) Left only (\( \infty \) never included)
\( (-\infty, \infty) \) All real numbers None (infinite)

Graphical Clues for Domain

★ Solid Dot
Endpoint is included in the domain. Use a bracket \( [ \) or \( ] \).
★ Open Dot
Endpoint is excluded. Use a parenthesis \( ( \) or \( ) \).
★ Arrow
Graph continues forever. Use \( \infty \) or \( -\infty \) with a parenthesis.
★ Vertical Asymptote
Graph never touches this \( x \)-value. Exclude it and use union \( \cup \).

Common Mistakes

Ignoring Context
Writing \( (-\infty, \infty) \) when the problem says "number of students." Students must be whole non-negative numbers.
Confusing Domain and Range
Reading the graph vertically (outputs) instead of horizontally (inputs). Domain is always about \( x \)-values.
Including Asymptotes
Using a bracket at a vertical asymptote. The function is undefined there: always use a parenthesis and union.

Question 1 of 5

What is the domain of \( f(x) = \sqrt{x - 4} \)?

Question 2 of 5

A function \( P(t) \) models a town's population where \( t \) is years since 2000. The model is valid through 2030. What is the most appropriate domain?

Question 3 of 5

The graph of a function has a solid dot at \( (-3, 1) \), an open dot at \( (5, 4) \), and no other gaps. What is the domain?

Question 4 of 5

What is the domain of \( g(x) = \dfrac{3}{x^2 - 9} \)?

Question 5 of 5

A ride-share company charges \( C(m) = 2.50m + 3 \) dollars for \( m \) miles. Trips are capped at 60 miles and must be at least 0.5 miles. What is the contextual domain?