Functions: Identity, Quadratic, Cubic & Absolute Value

Study Notes

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)

Description

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.