Functions & Their Representations

What is a Function?

Definition

A function f from a set S to a set Y is a rule that assigns a unique value y in Y to each value x in S.

• Domain: The set of all possible input values (x-values)
• Codomain: The set of all possible output values (y-values)

Notation

y = f(x) (read as “y is a function of x”)

The Four Ways to Represent a Function

1. Verbal (Words)

Describe the relationship in words

2. Numerical (Table)

Show values in a table

3. Algebraic (Formula)

Express as a mathematical formula

4. Visual (Graph)

Display on a coordinate graph

Real-World Examples

Example 1: Average Temperature

1️⃣ VERBAL “The average temperature of a city varies throughout the year, increasing from winter to summer and decreasing from summer to winter.”

2️⃣ NUMERICAL

3️⃣ VISUAL Graph shows temperature curve rising from winter to summer, then falling back to winter

Example 2: Distance and Speed

1️⃣ VERBAL A car travels at 60 miles per hour. The distance traveled depends on how many hours it has been driving.

2️⃣ ALGEBRAIC

d = 60t

Where:

• d = distance (miles)
• t = time (hours)
• 60 = speed (mph)

3️⃣ NUMERICAL

4️⃣ VISUAL Linear graph starting at origin (0,0) with slope of 60

Key Insight: The distance is a function of time. We can calculate distance for any time value using the formula d = 60t.

Example 3: Cell Phone Plans

VERBAL A cell phone plan costs $30/month plus $5 per GB of data used.

ALGEBRAIC

C = 30 + 5g

Where:

• C = total cost ($)
• g = data (GB)

NUMERICAL

VISUAL Linear graph starting at (0, 30) with slope of 5

Mapping Diagrams

A mapping diagram visually shows how elements from the domain map to the codomain using arrows.

Structure

Domain → Codomain
x₁ → f(x₁) = y₁
x₂ → f(x₂) = y₂
x₃ → f(x₃) = y₃

✓ The Rule of Functions

Each input must have exactly ONE output arrow

Example: f(x) = 2x

• 1 → 2
• 2 → 4
• 3 → 6

✓ This IS a function because each input has exactly one output

Key Takeaways

1. A function assigns exactly one output to each input
2. Functions can be represented four ways:
   • Verbally (in words)
   • Numerically (in tables)
   • Algebraically (as formulas)
   • Visually (as graphs)
3. The domain is the set of inputs; the codomain is the set of outputs
4. Notation: y = f(x) means “y is a function of x”
5. Different representations are useful in different situations

Practice Tip

Practice converting between all four representations to strengthen your understanding!

Additional Notes

• A function must pass the “vertical line test” on a graph - any vertical line should intersect the graph at most once
• In mapping diagrams, each element in the domain must have exactly one arrow pointing out
• Real-world applications help understand abstract function concepts