Introduction to Functions and
Domains
Mathematics Lesson for High School
/ Early College
Real-Life Analogy: Vending Machine
• Each input (coin) gives only one output
(snack). Just like a function!
Real-Life Analogy: Vending Machine
• ✅ 1. Vending Machine
• Input: Button pressed (e.g., A1, B2)
• Output: Selected snack
• Function? Yes. Each button press gives exactly
one product.
Real-Life Analogy: Student ID to
Student Name
• ✅ 2. Input: Student ID number
• Output: Student’s full name
• Function? Yes. One ID corresponds
to exactly one student.
Real-Life Analogy: 3. Days Since
Planting to Height of Plant
•
•
•
•
✅
Input: Number of days after planting
Output: Height of the plant in cm
Function? Yes. Each day has one
corresponding height measurement.
🎯 Objectives:
• Define a function.
• Identify if a relation is a
function.
• Determine the domain of a
function.
Definitions
• • Function: A relation where each input has
exactly one output.
• • Relation: A set of ordered pairs.
• • Domain: All possible input values (x-values).
• • Range: All possible output values (y-values).
Finding the Domain
• • From Set: List all x-values
• • From Graph: Look left to right
• • From Rule: Identify restrictions (e.g., no ÷0
or √(-))
Function vs. Not a Function
• • {(1, 2), (2, 3), (3, 4)} → Function
• • {(1, 2), (1, 3), (2, 4)} → Not a Function
• A function has only one output for each input.
Practice: Which Are Functions?
• Try identifying whether each relation is a
function.
• 1. {(2, 3), (3, 5), (2, 4)}
• 2. x → x²
• 3. Vertical Line x = 5
• 4. Equation: y = 2x + 1
Practice: Which Are Functions?
• 1. {(2, 3), (3, 5), (2, 4)}
• ❌ Not a Function
• Input 2 is associated with two different
outputs (3 and 4).
• Violates the rule that each input should map
to only one output.
Practice: Which Are Functions?
• 2. x → x²
• ✅ Function
• For any input x, you square it and get only one
output.
• Example: 2 → 4, –2 → 4 (note: same output is
fine; problem only if same input gives two
outputs).
• Passes the vertical line test.
Practice: Which Are Functions?
• 3. Vertical Line x = 5
• ❌ Not a Function
• A vertical line intersects the graph at many yvalues for a single x-value.
• Fails the vertical line test.
Practice: Which Are Functions?
• 4. Equation: y = 2x + 1
• ✅ Function
• Linear equation: each input x gives exactly one
output y.
• Passes vertical line test.
• This is the graph of a straight line (not
vertical), which is always a function.
Evaluate: Quick Quiz
• 1. {(1, 2), (2, 3), (3, 4)}
• 2. {(2, 4), (2, 5), (3, 6)}
• 3. x -> x + 3
• 4. x -> +/-sqrt(x)
• 5. A vertical line x = 7
Evaluate: Quick Quiz
1. {(1, 2), (2, 3), (3, 4)}
✅ Function
Each input (x-value) is unique and
maps to exactly one output.
✔ No x-value is repeated.
2. {(2, 4), (2, 5), (3, 6)}
❌ Not a Function
The input 2 is repeated with
two different outputs (4 and
5).
✘ One input gives more than
one output.
3. x → x + 3
✅ Function
This rule defines one output for every input x.
✔ For each x, there's only one result (x + 3).
4. x → ±√x
❌ Not a Function
This expression gives two
outputs (positive and
negative square roots) for
most x-values.
✘ One input → two outputs
→ not a function.
5. A vertical line x = 7
❌ Not a Function
A vertical line fails the Vertical Line Test because it has multiple outputs for a single xvalue.
✘ Not a function.
Evaluate: Quick Quiz
6. f(x) = x + 4
✅ Domain: All real numbers
📌 Reason: This is a linear
function. There are no square
roots or denominators to
restrict the domain.
Domain: x ∈ ℝ
9. f(x) = √(5 – x)
⛔ Restriction: 5 – x ≥ 0 → x ≤ 5
✅ Domain: All real numbers less than or equal
to 5
Domain: x ≤ 5
7. f(x) = 1 / (x - 2)
⛔ Restriction: Denominator ≠ 0 → x – 2 ≠ 0 →
x≠2
✅ Domain: All real numbers except 2
Domain: x ∈ ℝ, x ≠ 2
8. f(x) = √(x + 3)
⛔ Restriction: Expression inside √ must be ≥ 0
→ x + 3 ≥ 0 → x ≥ –3
✅ Domain: All real numbers greater than or
equal to –3
Domain: x ≥ –3
10. f(x) = 1 / √(x – 1)
⛔ Restrictions:
•Denominator ≠ 0 → √(x – 1) ≠ 0 →
x≠1
•Inside square root must be > 0
(not just ≥ 0 because it's in the
denominator):
x–1>0→x>1
✅ Domain: All real numbers
greater than 1
Domain: x > 1
Exit Ticket & Homework
• Exit Ticket:
• • One thing you learned
• • One question you still have
• Homework:
• • Find 3 real-life functions and list their
domains.