Domain and Range

What is Domain?

The domain of a function is the set of all possible input values (x-values).

Think of it as: All the x-values that "work" in the function

Example 1: f(x) = 2x + 3

Domain: All real numbers Any x-value works! (-∞, ∞)

Example 2: f(x) = 1/x

Domain: All real numbers except 0 Can't divide by zero! (-∞, 0) ∪ (0, ∞)

Example 3: f(x) = √x

Domain: x ≥ 0 Can't take square root of negative! [0, ∞)

What is Range?

The range of a function is the set of all possible output values (y-values).

Think of it as: All the y-values the function can produce

Example 1: f(x) = x²

Range: y ≥ 0 or [0, ∞) Squaring never gives negative result

Example 2: f(x) = 3

Range: {3} Constant function, only outputs 3

Example 3: f(x) = |x|

Range: y ≥ 0 or [0, ∞) Absolute value is never negative

Domain Restrictions

Common restrictions:

1. Division by zero If denominator can equal zero, exclude that x-value

Example: f(x) = 1/(x - 5)

x - 5 = 0 when x = 5 Domain: x ≠ 5 or (-∞, 5) ∪ (5, ∞)

2. Square roots (even roots) Expression under square root must be non-negative

Example: f(x) = √(x + 3)

x + 3 ≥ 0 x ≥ -3 Domain: [-3, ∞)

3. Logarithms (in advanced courses) Argument must be positive

4. Real-world constraints Time can't be negative, number of people must be whole numbers, etc.

Finding Domain from Equations

Step 1: Look for restrictions Step 2: Set up conditions Step 3: Solve inequalities Step 4: Write domain

Example 1: f(x) = 3x - 7

No restrictions! Domain: All real numbers or (-∞, ∞)

Example 2: f(x) = 1/(x + 2)

Restriction: x + 2 ≠ 0 x ≠ -2 Domain: (-∞, -2) ∪ (-2, ∞)

Example 3: f(x) = √(2x - 6)

Restriction: 2x - 6 ≥ 0 2x ≥ 6 x ≥ 3 Domain: [3, ∞)

Example 4: f(x) = 1/(x² - 9)

x² - 9 = 0 x² = 9 x = ±3

Domain: (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)

Example 5: f(x) = √(x - 1)/(x - 4)

Two restrictions:

1. x - 1 ≥ 0 → x ≥ 1
2. x - 4 ≠ 0 → x ≠ 4

Domain: [1, 4) ∪ (4, ∞)

Finding Range from Equations

For Algebra 1, often use graphing or logic

Linear functions: f(x) = mx + b (m ≠ 0) Range: All real numbers (-∞, ∞)

Example: f(x) = 2x + 1 Range: (-∞, ∞)

Quadratic functions: f(x) = ax² + bx + c

If a > 0 (opens up): Range: [minimum y-value, ∞)

If a < 0 (opens down): Range: (-∞, maximum y-value]

Example 1: f(x) = x² Opens up, vertex at (0, 0) Range: [0, ∞)

Example 2: f(x) = -x² + 4 Opens down, vertex at (0, 4) Range: (-∞, 4]

Example 3: f(x) = (x - 2)² + 1 Opens up, vertex at (2, 1) Range: [1, ∞)

Absolute value: f(x) = a|x - h| + k

If a > 0: Range: [k, ∞)

If a < 0: Range: (-∞, k]

Example: f(x) = |x| - 3 Range: [-3, ∞)

Interval Notation

Used to express domain and range concisely

Symbols:

• ( or ) : Does NOT include endpoint (open)
• [ or ] : INCLUDES endpoint (closed)
• ∞ : Always use ( or ) never [ or ]

Examples:

(3, 7) : 3 < x < 7 (3 and 7 not included) [3, 7] : 3 ≤ x ≤ 7 (3 and 7 included) [3, 7) : 3 ≤ x < 7 (3 included, 7 not) (-∞, 5] : x ≤ 5 (all numbers up to and including 5) [2, ∞) : x ≥ 2 (all numbers from 2 onward) (-∞, ∞) : All real numbers

Union symbol ∪: Combines separate intervals

Example: (-∞, 2) ∪ (2, ∞) All real numbers except 2

Example: [-3, 0) ∪ (0, 5] From -3 to 5, but excluding 0

Set Notation

Alternative to interval notation

Examples:

{x | x > 3} : "The set of all x such that x is greater than 3" {x | x ≠ 0} : "All real numbers except 0" {x | -2 ≤ x ≤ 5} : "All x between -2 and 5, inclusive"

For all real numbers: ℝ or {x | x ∈ ℝ}

Finding Domain and Range from Graphs

Domain: Look at x-values covered (left to right)

Range: Look at y-values covered (bottom to top)

Example 1: Line from (-2, 1) to (3, 4)

Domain: [-2, 3] (endpoints included if dots are solid) Range: [1, 4]

Example 2: Parabola y = x² - 4

Graph extends infinitely left and right Domain: (-∞, ∞)

Lowest point at y = -4, extends up infinitely Range: [-4, ∞)

Example 3: Horizontal line at y = 2

Extends infinitely left and right Domain: (-∞, ∞)

Only y-value is 2 Range: {2}

Example 4: Circle (not a function!)

If center (0, 0) and radius 3: Domain: [-3, 3] Range: [-3, 3]

Note: Circle fails vertical line test (not a function)

Domain and Range from Tables

Domain: List all x-values (inputs) Range: List all y-values (outputs)

Example:

x | y --|-- 1 | 5 2 | 7 3 | 9 4 | 11

Domain: {1, 2, 3, 4} Range: {5, 7, 9, 11}

Example 2:

x | y --|-- 0 | 3 1 | 3 2 | 3 3 | 3

Domain: {0, 1, 2, 3} Range: {3} (only one y-value!)

Common Function Types

Linear (y = mx + b): Domain: (-∞, ∞) Range: (-∞, ∞) if m ≠ 0 Range: {b} if m = 0 (horizontal line)

Quadratic (y = ax²): Domain: (-∞, ∞) Range: [0, ∞) if a > 0 Range: (-∞, 0] if a < 0

Absolute Value (y = |x|): Domain: (-∞, ∞) Range: [0, ∞)

Square Root (y = √x): Domain: [0, ∞) Range: [0, ∞)

Reciprocal (y = 1/x): Domain: (-∞, 0) ∪ (0, ∞) Range: (-∞, 0) ∪ (0, ∞)

Constant (y = c): Domain: (-∞, ∞) Range: {c}

Real-World Contexts

Example 1: Area of square

A(s) = s² where s = side length

Domain: s > 0 (can't have negative or zero side) Range: A > 0 (area is positive)

Example 2: Cost function

C(n) = 5n + 20 where n = number of items

Domain: n ≥ 0 (can't buy negative items) Often: n is whole number (0, 1, 2, 3, ...) Range: C ≥ 20 (minimum cost is 20)

Example 3: Projectile height

h(t) = -16t² + 64t where t = time in seconds

Domain: 0 ≤ t ≤ 4 (from launch until it hits ground) Range: 0 ≤ h ≤ 64 (ground to maximum height)

Example 4: Temperature over 24 hours

T(h) where h = hour (0 to 24)

Domain: [0, 24] Range: Depends on actual temperatures, maybe [45, 75]

Discrete vs Continuous

Discrete: Separate points (often integers)

Example: Number of students in class Domain: {0, 1, 2, 3, ..., 30} (whole numbers only)

Continuous: Unbroken interval

Example: Height of plant over time Domain: [0, ∞) (any non-negative time)

In Algebra 1: Often assume continuous unless context requires discrete

Determining if Value is in Domain

Substitute and check if result is defined

Example: Is x = 3 in domain of f(x) = 1/(x - 3)?

Try x = 3: f(3) = 1/(3 - 3) = 1/0 (undefined!)

No, x = 3 is NOT in domain

Example 2: Is x = -2 in domain of f(x) = √(x + 5)?

Try x = -2: f(-2) = √(-2 + 5) = √3 (defined!)

Yes, x = -2 IS in domain

Determining if Value is in Range

Check if there's an x-value that produces that y-value

Example: Is y = 4 in range of f(x) = x²?

Set 4 = x² x = ±2 (solutions exist!)

Yes, y = 4 IS in range

Example 2: Is y = -1 in range of f(x) = x²?

Set -1 = x² No real solutions!

No, y = -1 is NOT in range

Multiple Restrictions

Example: f(x) = √x / (x - 4)

Restriction 1: x ≥ 0 (square root) Restriction 2: x ≠ 4 (division by zero)

Combined: [0, 4) ∪ (4, ∞)

Must satisfy BOTH conditions!

Example 2: f(x) = √(9 - x²)

9 - x² ≥ 0 x² ≤ 9 -3 ≤ x ≤ 3

Domain: [-3, 3]

Common Mistakes to Avoid

1. Forgetting to check for division by zero Always set denominator ≠ 0!

2. Square root of negative Under square root must be ≥ 0

3. Confusing domain and range Domain = inputs (x), Range = outputs (y)

4. Wrong inequality direction √(x - 3) requires x - 3 ≥ 0, so x ≥ 3 (not x ≤ 3!)

5. Including infinity with square bracket ALWAYS use ∞ with ( or ) NEVER [∞ or ∞]

6. Not simplifying x² - 4 ≠ 0 means x ≠ ±2 (factor!)

7. Forgetting real-world constraints Time can't be negative, people must be whole numbers

Piecewise Functions

Different rules for different parts of domain

Example:

f(x) = {x + 1 if x < 0 {2x if x ≥ 0

Domain: All real numbers (-∞, ∞) Range: Depends on both pieces

For x < 0: outputs are less than 1 For x ≥ 0: outputs are ≥ 0

Combined range: (-∞, ∞)

Restricting Domain

Sometimes we artificially limit domain

Example: f(x) = x² with domain [-2, 3]

Even though x² works for all real numbers, we restrict to [-2, 3]

Range: [0, 9] (minimum at x = 0 gives y = 0, maximum at x = 3 gives y = 9)

Practical Applications

Domain considerations:

• Can time be negative? (usually no)
• Can quantity be fractional? (sometimes no)
• Are there physical limits? (height, weight, distance)
• Are there legal limits? (age, speed limit)

Range considerations:

• What are minimum/maximum values?
• Can output be negative?
• Are there practical limits?

Quick Reference

Domain: All possible x-values (inputs)

Range: All possible y-values (outputs)

Restrictions:

• Division by zero: denominator ≠ 0
• Square root: expression ≥ 0
• Real-world: context-dependent

Interval Notation:

• ( ) : not included
• [ ] : included
• ∪ : union (combine)
• ∞ : always with ( or )

All real numbers: (-∞, ∞)

From graph:

• Domain: look left to right
• Range: look bottom to top

Practice Strategy

• Always check for division by zero first
• Look for square roots (even roots)
• Consider real-world context
• Practice interval notation
• Graph to verify domain and range
• Test boundary values
• Check if endpoints included or excluded
• Work with various function types
• Understand discrete vs continuous
• Remember: domain is INPUT, range is OUTPUT
• Use test points to verify
• Draw number lines to visualize
• Master inequality solving
• Practice reading domains/ranges from graphs

Understanding domain and range is crucial for working with functions throughout algebra and beyond. These concepts appear in calculus, statistics, and real-world applications everywhere!