Questions and Answers
What is the identity function f(x)?
The quadratic function has a range of all real numbers.
False
What symmetry does the cubic function have?
Origin symmetry
What is the range of the absolute value function f(x)=|x|?
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What is another name for the greatest integer function?
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The exponential function f(x)=2^x touches the x-axis.
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What does the reciprocal function approach but never touches?
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The domain of the square root function f(x)=√x is __________.
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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)
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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.