1- Functions

Instructors:
- Ahmed El Sherif
  - Position: Assistant Professor – Engineering Mathematics
  - Department: Basic Science Department
  - Email: ahmed.elsherif@bue.edu.eg
- Mohamed Aboshady
  - Position: Assistant Professor – Engineering Mathematics
  - Department: Basic Science Department
  - Email: Mohamed.Aboshady@bue.edu.eg

Textbook: Calculus
- Authors: Robert T. Smith (Millersville University of Pennsylvania), Roland B. Minton (Roanoke College)

Real Number System & Inequalities:
- Theorem 1.1 Properties of inequalities:
  - If a < b, then:
    - For any real number C, atc < btc
    - For C < d, then a + C < b + d.
    - If C < 0, then aC > bC.
Example 1.1: Solving a Linear Inequality
- Solve 2x + 5 < 13:
  - Steps:
    1. Subtract 5:
      - 2x < 8
    2. Divide by 2:
      - x < 4
  - Solution in interval notation: (-∞, 4)
Example 1.2: Solving a Two-Sided Inequality
- Solve 6 < 1 - 3x < 10
Example 1.3: Solving an Inequality Involving a Fraction
- Solve (x - 1)/(x + 2) < 2
  - Solution: (-∞, -2) ∪ (1, ∞)
Example 1.4: Solving a Quadratic Inequality
- Solve x² + x - 6 > 0
  - Solution: (-∞, -3) ∪ (2, ∞)

**Definitions:
- The slope of a line through points (x1, y1) and (x2, y2):
  - m = (y2 - y1)/(x2 - x1)
- If x1 = x2 and y1 ≠ y2, the slope is undefined.
Example 2.1: Finding the Slope of a Line
- Through points (4, 3) and (2, 5):
  - m = (3 - 5)/(4 - 2) = -1
Theorem 2.1: Parallel and Perpendicular Lines
- Two lines are parallel if they have the same slope.
- Two lines are perpendicular if m1 * m2 = -1.
Definition 2.2: Function
- A rule that assigns one element y in set B for each element x in set A, denoted y = f(x).
- Set A is the domain; set B is the range.

Definition 4.1: Periodic Function
- A function f is periodic of period T if f(x + T) = f(x).
Basic Trigonometric Functions:
- Sine: f(θ) = sin(θ)
- Cosine: g(θ) = cos(θ)
- Both have a period of 2π.
Example 4.1: Solve Equations Involving Sines and Cosines
- (a) 2 sin(x) - 1 = 0 produces solutions for sin(x) = 1/2.
- (b) cos²(x) - 3 cos(x) + 2 = 0 gives cos(x) = 1, produces no solution for cos(x) = 2 as it's outside the range.

Definition 5.1: Combinations of Functions
Vertical Translation:
- For graph y = f(x) + C, shifts graph vertically by |C| units.
Horizontal Translation:
- For graph y = f(x + C), shifts graph left or right by C units depending on the sign.
Example 5.1: Finding the Composition of Two Functions
- For f(x) = x² + 1 and g(x) = x - 2, compositions depend on domains.