Domain and Range

What are Domain and Range?

Domain: The set of all possible INPUT values (x-values) for a function.

Range: The set of all possible OUTPUT values (y-values or f(x)-values) for a function.

Think of it this way:

• Domain = What you can PUT INTO the function
• Range = What you can GET OUT OF the function

Why Domain and Range Matter

Not all inputs make sense for every function:

• Can't divide by zero
• Can't take square root of negative (in real numbers)
• Real-world constraints (can't have negative time)

Understanding domain and range helps you:

• Know when a function is defined
• Understand function behavior
• Solve real-world problems correctly
• Avoid mathematical errors

Interval Notation

We use interval notation to write domain and range efficiently.

Symbols:

• (a, b) means all numbers between a and b, NOT including a or b (open interval)
• [a, b] means all numbers between a and b, INCLUDING both a and b (closed interval)
• (a, b] means between a and b, NOT including a but INCLUDING b
• [a, b) means between a and b, INCLUDING a but NOT including b
• ∞ (infinity) always uses parentheses: (a, ∞) or (-∞, b)

Examples:

• (2, 5) means 2 < x < 5
• [2, 5] means 2 ≤ x ≤ 5
• (-∞, 3] means x ≤ 3
• [4, ∞) means x ≥ 4
• (-∞, ∞) means all real numbers

Set Notation

Another way to express domain and range:

{x | condition} reads as "the set of all x such that condition"

Examples:

• {x | x > 0} means "all x greater than 0"
• {x | x ≠ 2} means "all x except 2"
• {x | x ∈ ℝ} means "all real numbers"

Domain of Linear Functions

Linear functions: f(x) = mx + b

Domain: ALL real numbers Written as: (-∞, ∞) or {x | x ∈ ℝ}

Why? You can plug any number into a linear function.

Examples:

• f(x) = 2x + 3, Domain: (-∞, ∞)
• g(x) = -x + 7, Domain: (-∞, ∞)
• h(x) = 5, Domain: (-∞, ∞)

Range of Linear Functions

Non-constant linear: Range is ALL real numbers (-∞, ∞)

Constant function: f(x) = c Range is just {c} (a single value)

Examples:

• f(x) = 2x + 3, Range: (-∞, ∞)
• g(x) = 5, Range: {5} or [5, 5]

Domain of Quadratic Functions

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

Domain: ALL real numbers (-∞, ∞)

Why? You can square any real number.

Examples:

• f(x) = x², Domain: (-∞, ∞)
• g(x) = -2x² + 3x - 1, Domain: (-∞, ∞)

Range of Quadratic Functions

Range depends on the vertex and direction of opening.

If parabola opens UP (a > 0): Range: [k, ∞) where k is the y-coordinate of the vertex (minimum value)

If parabola opens DOWN (a < 0): Range: (-∞, k] where k is the y-coordinate of the vertex (maximum value)

Example 1: f(x) = x² - 4x + 3

Find vertex: x = -(-4)/(2·1) = 2 f(2) = 4 - 8 + 3 = -1 Vertex: (2, -1)

Opens up (a = 1 > 0) Range: [-1, ∞)

Example 2: g(x) = -x² + 6x - 5

Vertex: x = -6/(2·(-1)) = 3 g(3) = -9 + 18 - 5 = 4 Vertex: (3, 4)

Opens down (a = -1 < 0) Range: (-∞, 4]

Domain Restrictions: Division

Cannot divide by zero!

When a function has a variable in the denominator, exclude values that make the denominator zero.

Example 1: f(x) = 1/x

Denominator = x Set equal to zero: x = 0 Domain: All real numbers EXCEPT 0 Written as: (-∞, 0) ∪ (0, ∞) or {x | x ≠ 0}

Example 2: g(x) = 3/(x - 5)

Denominator = x - 5 Set equal to zero: x - 5 = 0, so x = 5 Domain: All real numbers EXCEPT 5 Written as: (-∞, 5) ∪ (5, ∞) or {x | x ≠ 5}

Example 3: h(x) = (2x + 1)/(x² - 4)

Denominator = x² - 4 Set equal to zero: x² - 4 = 0 x² = 4 x = ±2

Domain: All real numbers EXCEPT 2 and -2 Written as: (-∞, -2) ∪ (-2, 2) ∪ (2, ∞) or {x | x ≠ ±2}

Domain Restrictions: Square Roots

Cannot take square root of negative number (in real numbers).

When a function has a square root, the expression inside must be ≥ 0.

Example 1: f(x) = √x

Inside square root: x Requirement: x ≥ 0 Domain: [0, ∞)

Example 2: g(x) = √(x - 3)

Inside: x - 3 Requirement: x - 3 ≥ 0 Solve: x ≥ 3 Domain: [3, ∞)

Example 3: h(x) = √(5 - x)

Inside: 5 - x Requirement: 5 - x ≥ 0 Solve: -x ≥ -5 Multiply by -1 (flip inequality): x ≤ 5 Domain: (-∞, 5]

Example 4: f(x) = √(x² - 9)

Inside: x² - 9 Requirement: x² - 9 ≥ 0 x² ≥ 9 |x| ≥ 3 This means x ≤ -3 or x ≥ 3 Domain: (-∞, -3] ∪ [3, ∞)

Combined Restrictions

Sometimes functions have BOTH division and square roots.

Example: f(x) = √x / (x - 4)

Square root restriction: x ≥ 0 Division restriction: x ≠ 4

Combined domain: [0, 4) ∪ (4, ∞)

We need x ≥ 0 AND x ≠ 4.

Finding Range from Graphs

From a graph:

1. Look at the y-axis
2. Identify the lowest and highest y-values the graph reaches
3. Write the range

Example: If a graph goes from y = -2 up to y = 5 (including both) Range: [-2, 5]

Example: If a parabola opening up has vertex at y = 3 Range: [3, ∞)

Example: If a parabola opening down has vertex at y = -1 Range: (-∞, -1]

Finding Domain from Graphs

From a graph:

1. Look at the x-axis
2. Identify all x-values that have corresponding points on the graph
3. Write the domain

Example: If a graph extends forever left and right Domain: (-∞, ∞)

Example: If a graph starts at x = 2 and extends right Domain: [2, ∞)

Example: If there's a gap (like at x = 3) Domain might be: (-∞, 3) ∪ (3, ∞)

Absolute Value Functions

f(x) = |x| Domain: (-∞, ∞) Range: [0, ∞) (absolute value is never negative)

f(x) = |x - h| + k Domain: (-∞, ∞) Range: [k, ∞) if upright V, (-∞, k] if upside-down

Example: f(x) = |x - 2| + 3 Domain: (-∞, ∞) Range: [3, ∞) The vertex of the V is at (2, 3).

Piecewise Functions

Domain is typically all real numbers (unless specified). Range depends on the individual pieces.

Example: f(x) = x + 1 if x < 0 f(x) = x² if x ≥ 0

Domain: (-∞, ∞)

For range, consider both pieces:

• When x < 0: f(x) = x + 1 gives values from (-∞, 1)
• When x ≥ 0: f(x) = x² gives values from [0, ∞)

Range: (-∞, 1) ∪ [0, ∞) = (-∞, ∞)

Real-World Domain and Range

Context matters! Physical constraints restrict domain and range.

Example 1: Area of square A(s) = s² where s is side length

Mathematical domain: (-∞, ∞) Real-world domain: (0, ∞) because side length must be positive

Mathematical range: [0, ∞) Real-world range: (0, ∞) because area must be positive

Example 2: Projectile h(t) = -16t² + 64t where t is time

Real-world domain: [0, 4] (from launch until it hits ground) Real-world range: [0, 64] (from ground to maximum height)

Example 3: Cost function C(n) = 50 + 10n where n is number of items

Real-world domain: {0, 1, 2, 3, ...} (can't buy 2.5 items) Real-world range: {50, 60, 70, 80, ...} (discrete values)

Practice: Finding Domain

For each function, find the domain:

Problem 1: f(x) = 3x - 7 Linear, no restrictions Domain: (-∞, ∞)

Problem 2: g(x) = 1/(x + 2) Denominator zero when x = -2 Domain: (-∞, -2) ∪ (-2, ∞)

Problem 3: h(x) = √(x + 4) Inside must be ≥ 0: x + 4 ≥ 0, so x ≥ -4 Domain: [-4, ∞)

Problem 4: f(x) = x² No restrictions Domain: (-∞, ∞)

Problem 5: g(x) = √(2x - 6) Inside ≥ 0: 2x - 6 ≥ 0, so 2x ≥ 6, thus x ≥ 3 Domain: [3, ∞)

Practice: Finding Range

Problem 1: f(x) = x - 5 Linear (non-constant) Range: (-∞, ∞)

Problem 2: g(x) = x² + 1 Parabola opening up, vertex at (0, 1) Range: [1, ∞)

Problem 3: h(x) = -x² + 4 Parabola opening down, vertex at (0, 4) Range: (-∞, 4]

Problem 4: f(x) = |x| - 3 V-shape with vertex at (0, -3), opens up Range: [-3, ∞)

Problem 5: g(x) = 7 Constant function Range: {7}

Union Notation

Use ∪ (union) to combine separate intervals.

Example: x < 2 or x > 5 Written as: (-∞, 2) ∪ (5, ∞)

Example: All real numbers except x = 3 Written as: (-∞, 3) ∪ (3, ∞)

Example: -4 ≤ x < -1 or 2 < x ≤ 7 Written as: [-4, -1) ∪ (2, 7]

Common Mistakes to Avoid

1. Using wrong brackets Remember: ( ) for open, [ ] for closed Infinity ALWAYS uses ( )

2. Forgetting to solve inequalities √(x - 3) requires x - 3 ≥ 0, which gives x ≥ 3

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

4. Missing restrictions Always check for division by zero and square roots

5. Ignoring graph direction Parabola up: range [k, ∞) Parabola down: range (-∞, k]

6. Writing infinity with brackets Never write [∞) or (-∞] Always (∞) and (-∞)

Domain and Range from Tables

Table:

Domain: {-2, 0, 2, 4} (discrete values shown) Range: {3, 5, 7, 9}

If the pattern continues forever: Domain: all even integers Range: all odd integers starting from 3

Continuous vs. Discrete

Continuous domain/range: Any value in an interval Example: (-∞, ∞), [2, 5]

Discrete domain/range: Only specific separate values Example: {0, 1, 2, 3, ...}, {-2, 0, 2, 4}

Real-world examples of discrete:

• Number of students (can't have 25.5 students)
• Number of cars (whole numbers only)
• Days of the week (discrete set)

Testing Your Understanding

Quick checks:

For domain:

• Can I divide by zero? If yes, exclude that x-value
• Is there a square root? If yes, set inside ≥ 0
• Are there real-world constraints? If yes, apply them

For range:

• Is it linear (non-constant)? Range is (-∞, ∞)
• Is it quadratic? Find vertex, check if up or down
• Is there a minimum or maximum? Use [ or ]
• Can I reach all y-values or only some?

Step-by-Step Process

Finding Domain:

1. Identify function type
2. Check for division (denominator ≠ 0)
3. Check for square roots (inside ≥ 0)
4. Check for other restrictions
5. Apply real-world constraints if applicable
6. Write in interval notation

Finding Range:

1. Determine function type
2. For linear: usually (-∞, ∞) unless constant
3. For quadratic: find vertex and direction
4. For other functions: analyze graph or behavior
5. Apply real-world constraints if applicable
6. Write in interval notation

Advanced Example

f(x) = √(4 - x²)

Domain: Inside square root: 4 - x² ≥ 0 -x² ≥ -4 x² ≤ 4 |x| ≤ 2 -2 ≤ x ≤ 2 Domain: [-2, 2]

Range: When x = 0: f(0) = √4 = 2 (maximum) When x = ±2: f(±2) = √0 = 0 (minimum) Range: [0, 2]

This is actually the upper half of a circle!

Quick Reference

Common Domains:

• Linear: (-∞, ∞)
• Quadratic: (-∞, ∞)
• f(x) = 1/x: (-∞, 0) ∪ (0, ∞)
• f(x) = √x: [0, ∞)
• Absolute value: (-∞, ∞)

Common Ranges:

• Linear (non-constant): (-∞, ∞)
• Constant: {c}
• Quadratic up: [k, ∞)
• Quadratic down: (-∞, k]
• f(x) = √x: [0, ∞)
• f(x) = |x|: [0, ∞)

Tips for Success

• Always check for division by zero
• Square roots need non-negative inputs
• Graph the function if unsure about range
• Use correct bracket notation
• Consider real-world constraints
• Domain comes from input restrictions
• Range comes from output possibilities
• Practice with various function types
• When in doubt, make a table or graph