Essential Mathematical Formulas: Sets, Functions, and Trigonometry

Properties of Sets

Important Set Formulas

• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• If A and B are disjoint: n(A ∪ B) = n(A) + n(B)
• Number of subsets of a set with n elements = 2ⁿ
• Number of proper subsets = 2ⁿ – 1
• n(P(A)) = 2ⁿ

1. Identity Function

f(x) = x

• Graph: Straight line through origin
• Line: y = x
• Domain & Range: ℝ
• Passes through: (0, 0), (1, 1), (-1, -1)

2. Constant Function

f(x) = c

• Graph: Horizontal line
• Example: f(x) = 3
• Domain: ℝ
• Range: {c}

3. Linear Function

f(x) = ax + b

• Graph: Straight line
• Slope: a
• Intercepts: x = -b/a, y = b

4. Modulus Function

f(x) = |x|

• Graph: V-shaped
• Domain: ℝ
• Range: [0, ∞)
• Turns at: (0, 0)

5. Greatest Integer Function

f(x) = ⌊x⌋

• Graph: Step function
• Jumps at integers
• Domain: ℝ
• Range: All integers (ℤ)

6. Signum Function

f(x) = sign(x)

• f(x) = -1 when x < 0
• f(x) = 0 when x = 0
• f(x) = 1 when x > 0
• Domain: ℝ
• Range: {-1, 0, 1}

7. Square Function

f(x) = x²

• Graph: U-shaped parabola
• Vertex: (0, 0)
• Domain: ℝ
• Range: [0, ∞)

8. Cube Function

f(x) = x³

• Graph: S-shaped curve
• Passes through: (0, 0)
• Domain & Range: ℝ

9. Square Root Function

f(x) = √x

• Graph: Half-curve (right side)
• Starts from: (0, 0)
• Domain: [0, ∞)
• Range: [0, ∞)

10. Reciprocal Function

f(x) = 1/x

• Graph: Two curves in I and III quadrant
• x ≠ 0
• Domain: ℝ – {0}
• Range: ℝ – {0}
• Asymptotes: x = 0, y = 0

11. Exponential Function

f(x) = aˣ (a > 1)

• Passes through (0, 1)
• Grows rapidly
• Domain: ℝ
• Range: (0, ∞)

12. Logarithmic Function

f(x) = logₐx (a > 1)

• Passes through (1, 0)
• Domain: (0, ∞)
• Range: ℝ
• Vertical asymptote at x = 0

Radian Measures

Conversion Formulas:

• Degree to radian: θ (in rad) = (π/180) × θ°
• Radian to degree: θ° = (180/π) × θ (in rad)

Basic Trigonometric Identities

Trigonometric Table (Standard Angles)