BCA TU, Math I – Unit 2: Relation, Functions and Graphs 8 Hrs Exam Notes

1. Ordered Pairs and Cartesian Product

• Ordered Pair (a, b): A pair of elements where order matters (i.e., (a, b) ≠ (b, a) unless a = b).
• Example: (2, 3) ≠ (3, 2).
• Cartesian Product (A × B): Set of all ordered pairs (a, b), where a ∈ A and b ∈ B.
  • Formula: A × B = {(a, b) | a ∈ A, b ∈ B}.
  • Example: If A = {1, 2} and B = {a, b}, then A × B = {(1, a), (1, b), (2, a), (2, b)}.
• Graphical Representation: Cartesian plane (x, y coordinates).

2. Relations

• Definition: A relation R from set A to B is a subset of A × B.
• Example: If A = {1, 2, 3} and B = {4, 5}, then R = {(1,4), (2,5)} is a relation.

Domain and Range of a Relation

• Domain: Set of all first elements in relation (inputs).
• Range: Set of all second elements (outputs).
• Example: If R = {(1, 4), (2, 5), (3, 4)},
  • Domain = {1, 2, 3}
  • Range = {4, 5}.

Inverse of a Relation

• Definition: If R = {(a, b)}, then the inverse relation R⁻¹ = {(b, a)}.
• Example: If R = {(1, 4), (2, 5)}, then R⁻¹ = {(4, 1), (5, 2)}.

3. Types of Relations

• Reflexive:(a, a) ∈ R for all a ∈ A.
  • Example: R = {(1,1), (2,2), (3,3)} is reflexive in {1,2,3}.
• Symmetric: If (a, b) ∈ R, then (b, a) ∈ R.
  • Example: R = {(1,2), (2,1)} is symmetric.
• Transitive: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
  • Example: If R = {(1,2), (2,3), (1,3)}, it is transitive.
• Equivalence Relation: If R is Reflexive, Symmetric, and Transitive, it is an equivalence relation.
  • Example: R = {(1,1), (2,2), (3,3), (1,3), (3,1)}.

4. Functions

Definition of Function

• A function f: A → B assigns exactly one output f(a) to each input a from A.
• Example: f(x) = x², where input 2 gives f(2) = 4.

Domain and Range of a Function

• Domain: The set of all input values.
• Range: The set of output values.
• Example: If f(x) = x², then:
  • Domain: (-∞, ∞)
  • Range: [0, ∞).

Inverse Function

• Definition: If f: A → B has an inverse f⁻¹: B → A, then f(f⁻¹(x)) = x.
• Example: If f(x) = 2x + 3, its inverse is f⁻¹(x) = (x - 3)/2.
• Graphical Property: The graph of f⁻¹(x) is a reflection of f(x) across the line y = x.

5. Special Functions

1. Identity Function:f(x) = x.
   • Graph: Straight line passing through (0,0).
2. Constant Function:f(x) = c, where c is constant.
   • Graph: Horizontal line y = c.

6. Algebraic Functions and Graphs

1. Linear Function:f(x) = ax + b.
   • Example: f(x) = 2x + 1.
   • Graph: Straight line.
2. Quadratic Function:f(x) = ax² + bx + c.
   • Example: f(x) = x² - 4.
   • Graph: Parabola.
3. Cubic Function:f(x) = ax³ + bx² + cx + d.
   • Example: f(x) = x³ - x.
   • Graph: S-shaped curve.

7. Trigonometric Functions and Graphs

• Sine Function:f(x) = sin(x).
  • Graph: Wave oscillating between -1 and 1.
• Cosine Function:f(x) = cos(x).
  • Graph: Similar to sin(x), but shifted left.
• Tangent Function:f(x) = tan(x).
  • Graph: Has vertical asymptotes at x = ±π/2.

8. Exponential and Logarithmic Functions

Exponential Function

• Definition: f(x) = a^x, where a > 0.
• Example: f(x) = 2^x.
• Graph:
  • Always positive (f(x) > 0).
  • Passes through (0,1).
  • Increases exponentially for x > 0.

Logarithmic Function

• Definition: f(x) = log_a(x), inverse of exponential function.
• Example: f(x) = log_2(x).
• Graph:
  • Passes through (1,0).
  • Defined only for x > 0.
  • Increases but at a slower rate than exponential.

9. Composite Function

• Definition: If f: A → B and g: B → C, then the composite function (g ∘ f): A → C is given by:
  • (g ∘ f)(x) = g(f(x)).
• Example: If f(x) = x² + 1 and g(x) = 2x, then:
  • (g ∘ f)(x) = g(f(x)) = 2(x² + 1) = 2x² + 2.