Functions: Identity, Quadratic, Cubic & Absolute Value

Identity Function

• Defined as f(x) = x
• Domain and Range: (-∞, ∞); includes all real numbers
• X and Y intercepts at zero
• Characteristics: continuous, increasing across entire domain
• Exhibits origin symmetry and is classified as an odd function
• No extrema or asymptotes present

Quadratic Function

• Expressed as f(x) = x²
• Domain: (-∞, ∞); all real numbers, Range: [0, ∞), starts from zero
• X and Y intercepts located at zero
• Features: continuous, decreasing on (-∞, 0) and increasing on (0, ∞)
• Exhibits Y-axis symmetry, classified as an even function
• Minimum point at (0,0) with no asymptotes

Cubic Functions

• Formula: f(x) = x³
• Domain and Range: (-∞, ∞); covers all real numbers
• X and Y intercepts at zero
• Continuous and increasing over the entire domain
• Displays origin symmetry and is classified as an odd function
• No extrema or asymptotes

Absolute Value Functions

• Defined by f(x) = |x|
• Domain: (-∞, ∞); all real numbers, Range: [0, ∞)
• X and Y intercepts both at zero
• Characteristics: continuous, decreasing on (-∞, 0) and increasing on (0, ∞)
• Exhibits Y-axis symmetry, classified as an even function
• Minimum point at (0,0), lacking any asymptotes

Greatest Integer Function

• Known as the "step function", represented as f(x) = [x]
• Domain: (-∞, ∞); all real numbers, Range: set of integers
• X intercept ranges from [0, 1), with Y intercept at zero
• Discontinuous; increasing at every integer step
• Lacks symmetry, odd/even classification, extrema, and asymptotes

Exponent Function

• Made up of f(x) = 2^x
• Never intersects either axis
• Domain: (-∞, ∞); includes all real numbers, Range: (0, ∞)
• No X intercepts, Y intercept at 1
• Continuous and increasing across the entire domain
• Lacks symmetry, extrema, odd/even classification, and asymptotes

Reciprocal Function

• Formulated as f(x) = 1/x
• Approaches zero but never touches it, characterized by curvy lines
• Domain: x ≠ 0; Range: x ≠ 0
• No X or Y intercepts
• Discontinuous, decreasing on (-∞, 0) and (0, ∞)
• Exhibits origin symmetry and classified as an odd function
• Contains asymptotes: horizontal (y=0) and vertical (x=0)

Square Root Function

• Expressed as f(x) = √x
• Domain: [0, ∞); Range: [0, ∞)
• X and Y intercepts both at zero
• Continuous and increasing from [0, ∞)
• No symmetry or odd/even classification, and lacks asymptotes
• Contains a minimum point at (0,0)