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)

Explore the characteristics and definitions of various functions including identity, quadratic, cubic, and absolute value functions. Test your understanding of their domains, ranges, intercepts, and classifications such as even and odd functions. This quiz is perfect for students looking to solidify their grasp on foundational mathematical concepts.