Basic Functions Characteristics Quiz

Identity (Linear) Function

• Domain: All real numbers (-∞, ∞)
• Range: All real numbers (-∞, ∞)
• Continuous with no breaks
• Always increasing across its domain
• Symmetric with respect to the origin (odd symmetry)
• Unbounded in both directions
• No local extrema present
• No horizontal or vertical asymptotes

Squaring (Quadratic) Function

• Domain: All real numbers (-∞, ∞)
• Range: Non-negative real numbers [0, ∞)
• Continuous and unbroken
• Decreases on (-∞, 0) and increases on (0, ∞)
• Symmetric about the y-axis (even symmetry)
• Bounded below by 0
• Local minimum at (0, 0) with no maximum
• No horizontal or vertical asymptotes

Cubing Function

• Domain: All real numbers (-∞, ∞)
• Range: All real numbers (-∞, ∞)
• Continuous with no interruptions
• Always increasing throughout the entire domain
• Exhibits odd symmetry
• Unbounded in both directions
• No local extrema present
• No horizontal or vertical asymptotes

Square Root Function

• Domain: Non-negative real numbers [0, ∞)
• Range: Non-negative real numbers [0, ∞)
• Continuous and without breaks
• Increasing from (0, ∞)
• No symmetry
• Bounded below by 0
• Local minimum at (0, 0) with no maximum
• No horizontal or vertical asymptotes

Natural Logarithm Function

• Domain: Positive real numbers (0, ∞)
• Range: All real numbers (-∞, ∞)
• Continuous function
• Increasing on (0, ∞)
• No symmetry
• Unbounded in both directions
• No local extrema present
• No horizontal asymptotes; vertical asymptote at x=0

Reciprocal Function

• Domain: All real numbers except zero (-∞, 0) ∪ (0, ∞)
• Range: All real numbers except zero (-∞, 0)
• Contains infinite discontinuities at x=0
• Decreasing from (-∞, 0) and from (0, ∞)
• Symmetric with respect to the origin (odd symmetry)
• Unbounded in both directions
• No local extrema
• Horizontal asymptote at y=0; vertical asymptote at x=0

Exponential Function

• Domain: All real numbers (-∞, ∞)
• Range: Positive real numbers (0, ∞)
• Continuous function without interruptions
• Always increasing over its entire domain
• Lacks symmetry
• Bounded below by 0
• No local extrema present
• Horizontal asymptote at y=0

Sine Function

• Domain: All real numbers (-∞, ∞)
• Range: Values between -1 and 1 inclusive [-1, 1]
• Continuous, exhibiting periodic behavior
• Alternates between increasing and decreasing in φ (periodic waves)
• Symmetric with respect to the origin (odd symmetry)
• Bounded within the interval [-1, 1]
• Local maximum at (φ, 1) and minimum at (φ, -1)
• No horizontal or vertical asymptotes

Cosine Function

• Domain: All real numbers (-∞, ∞)
• Range: Values between -1 and 1 inclusive [-1, 1]
• Continuous with periodic behavior
• Alternates between increasing and decreasing in φ (periodic waves)
• Symmetric about the y-axis (even symmetry)
• Bounded within the interval [-1, 1]
• Local maximum at (φ, 1) and minimum at (φ, -1)
• No horizontal or vertical asymptotes

Greatest Integer Function

• Domain: All real numbers (-∞, ∞)
• Range: All integers
• Exhibits jump discontinuities
• Always increases (pieces) across its domain
• No symmetry present
• Unbounded in both directions
• No local extrema
• No horizontal or vertical asymptotes

Absolute Value Function

• Domain: All real numbers (-∞, ∞)
• Range: Non-negative real numbers [0, ∞)
• Continuous with no breaks
• Decreases on (-∞, 0) and increases on (0, ∞)
• Symmetric about the y-axis (even symmetry)
• Bounded below by 0
• Local minimum at (0, 0) with no maximum
• No horizontal or vertical asymptotes

Logistic Function

• Domain: All real numbers (-∞, ∞)
• Range: Values between 0 and 1 (0, 1)
• Continuous function
• Always increasing across its domain
• No symmetry present
• Bounded within the interval (0, 1)
• No local extrema
• Horizontal asymptotes at y=0 and y=1