Lesson 5 • 40 min read
Functions & Geometry
In This Lesson
1Relations vs Functions
Relation
Any set of ordered pairs (x, y). A relation connects inputs to outputs.
Example:
{(1, 2), (2, 4), (1, 5), (3, 6)}
Note: x=1 maps to both 2 and 5
Function
A special relation where each input (x) has exactly ONE output (y).
Example:
{(1, 2), (2, 4), (3, 5), (4, 6)}
Each x has only one y value
Key Difference:
All functions are relations, but not all relations are functions.
A function is like a vending machine: one button (input) gives one specific item (output). You can't press one button and get two different items!
2Vertical Line Test
The Vertical Line Test
If any vertical line crosses a graph more than once, it is NOT a function.
✓ IS a Function
- • Straight lines (not vertical)
- • Parabolas opening up/down (y = x²)
- • Exponential curves
Every vertical line crosses only once
✗ NOT a Function
- • Circles (x² + y² = r²)
- • Vertical lines (x = 3)
- • Sideways parabolas (x = y²)
Some vertical lines cross more than once
Why Does This Work?
A vertical line represents all points with the same x-value. If it crosses the graph twice, that means one x-value produces two different y-values — violating the function definition!
3Domain & Range
Domain
All possible INPUT values (x-values)
"What can I put IN?"
Range
All possible OUTPUT values (y-values)
"What can come OUT?"
Example 1: From Ordered Pairs
{(1, 4), (2, 5), (3, 6), (4, 7)}
Domain: {1, 2, 3, 4}
Range: {4, 5, 6, 7}
Example 2: From a Graph
For a line from point (0, 2) to point (5, 7):
Domain: 0 ≤ x ≤ 5 (x goes from 0 to 5)
Range: 2 ≤ y ≤ 7 (y goes from 2 to 7)
Example 3: From an Equation
y = x²
Domain: All real numbers (any x works)
Range: y ≥ 0 (squares are never negative)
Domain Restrictions
- • Fractions: Denominator ≠ 0
- • Square roots: Expression under root ≥ 0
- • Real-world: Context limits (can't have negative people)
4Triangle Congruence
Two triangles are congruent if they have the same size and shape (all corresponding sides and angles are equal). Here are the postulates to prove congruence:
SSS (Side-Side-Side)
All three sides of one triangle equal the three sides of another
AB = DE, BC = EF, AC = DF
SAS (Side-Angle-Side)
Two sides and the included angle are equal
AB = DE, ∠B = ∠E, BC = EF
ASA (Angle-Side-Angle)
Two angles and the included side are equal
∠A = ∠D, AB = DE, ∠B = ∠E
AAS (Angle-Angle-Side)
Two angles and a non-included side are equal
∠A = ∠D, ∠B = ∠E, BC = EF
HL (Hypotenuse-Leg)
For RIGHT triangles only: If the hypotenuse and one leg are equal
⚠️ NOT Valid Congruence Tests
AAA (Angle-Angle-Angle)
Only proves similarity, not congruence
SSA (Side-Side-Angle)
The "ambiguous case" - may have 0, 1, or 2 solutions
5Basic Probability
Probability Formula
P(Event) = Favorable Outcomes / Total Outcomes
Probability is always between 0 and 1 (or 0% to 100%)
0
Impossible
0.5
Equally Likely
1
Certain
Example 1: Rolling a Die
P(rolling a 4): 1/6 (one 4 out of six sides)
P(rolling even): 3/6 = 1/2 (2, 4, 6 are even)
P(rolling < 5): 4/6 = 2/3 (1, 2, 3, 4 are less than 5)
Example 2: Drawing Cards
From a standard deck of 52 cards:
P(heart): 13/52 = 1/4
P(king): 4/52 = 1/13
P(red card): 26/52 = 1/2
Important Concepts
Complement: P(not A) = 1 - P(A)
Example: P(not rolling 6) = 1 - 1/6 = 5/6
Sample Space: All possible outcomes
Example: Coin flip = {H, T}, Die = {1,2,3,4,5,6}