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Questions and Answers
Which of the following describes a continuous function?
Which of the following describes a continuous function?
- A function that only increases
- A function with breaks in its domain
- A function that only decreases
- A function without holes or breaks (correct)
A vertical asymptote is a line that a curve intersects.
A vertical asymptote is a line that a curve intersects.
False (B)
What is the definition of an inflection point?
What is the definition of an inflection point?
A point where the concavity of the function changes.
The x-intercept occurs when the point is (______ , 0).
The x-intercept occurs when the point is (______ , 0).
What is an example of a discontinuous function?
What is an example of a discontinuous function?
Match the function type with its characteristic:
Match the function type with its characteristic:
Turning points indicate where the slope changes from increasing to decreasing.
Turning points indicate where the slope changes from increasing to decreasing.
What are global maxima and minima?
What are global maxima and minima?
Which of the following statements defines a function?
Which of the following statements defines a function?
The vertical line test can be used to determine if a graph represents a function.
The vertical line test can be used to determine if a graph represents a function.
What is the graphical representation of the function $y = |x|$?
What is the graphical representation of the function $y = |x|$?
The set of all possible input values for a function is called the ______.
The set of all possible input values for a function is called the ______.
For which of the following equations does the vertical line test apply?
For which of the following equations does the vertical line test apply?
Match the following types of functions with their characteristics:
Match the following types of functions with their characteristics:
In the relation {(1,2), (1,3), (2,4)}, the value '1' is considered the ______.
In the relation {(1,2), (1,3), (2,4)}, the value '1' is considered the ______.
What does the range of a function represent?
What does the range of a function represent?
Which of the following equations represents a polynomial function?
Which of the following equations represents a polynomial function?
The range of a polynomial function is always limited.
The range of a polynomial function is always limited.
What is the leading coefficient in the polynomial function y = -5x^4 + x^3 + 1?
What is the leading coefficient in the polynomial function y = -5x^4 + x^3 + 1?
A polynomial function can have _____ as its domain.
A polynomial function can have _____ as its domain.
Match the type of function with its characteristic:
Match the type of function with its characteristic:
What does 'n' indicate in the standard form of a polynomial expression?
What does 'n' indicate in the standard form of a polynomial expression?
The vertical line test can be used to determine if a relation is a function.
The vertical line test can be used to determine if a relation is a function.
Identify the constant term in the polynomial function f(x) = 3x^4 - 2x^2 + 5.
Identify the constant term in the polynomial function f(x) = 3x^4 - 2x^2 + 5.
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Study Notes
Continuous Functions
- Functions without holes or breaks across their entire domain.
- Types include Constant, Linear, Quadratic, Cubic, and Absolute Value.
Discontinuous Functions
- Functions containing at least one hole or break in their domain.
- Examples:
- Domain: 𝑥 ϵ (− ∞, 2) ∪ (2, + ∞); Range: 𝑦 ϵ (− ∞, 7) ∪ (7, + ∞)
- Domain: 𝑥 ϵ (− ∞, 3] ∪ (3, + ∞); Range: 𝑦 ϵ (− ∞, 1] ∪ (4, + ∞)
- Domain: 𝑥 ϵ (− ∞, 3) ∪ (3, + ∞); Range: 𝑦 ϵ (− ∞, 0) ∪ (0, + ∞)
Asymptote
- Lines (vertical, horizontal, oblique) that a curve approaches but never intersects.
Intercepts
- X-intercept: Occurs when y = 0 at the point (x, 0).
- Y-intercept: Occurs when x = 0 at the point (0, y).
Turning Points
- Points on a graph where the slope changes direction.
- Minimum points indicate a transition from decreasing to increasing interval.
- Maximum points indicate a transition from increasing to decreasing interval.
Inflection Points
- Locations where the concavity of the function changes.
- Concave Down: A line segment joining two points lies below the curve.
- Concave Up: A line segment joining two points lies above the curve.
Extrema Points
- Global Maxima: The highest point in the entire domain.
- Global Minima: The lowest point in the entire domain.
- Polynomial functions can be unbounded.
Polynomial Functions
- Defined with standard forms: n is a non-negative integer, x is the variable, an are real coefficients.
- Leading coefficient (an) is the coefficient of the term with the highest degree.
- Domain: Real numbers (𝐷 = {𝑥 | 𝑥 ϵ 𝑅}).
- Range can be all real numbers or have bounds.
Non-Polynomial Functions
- Some relations are functions but not polynomial:
- Trigonometric function (e.g., 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥)
- Exponential function (e.g., 𝑔(𝑥) = 𝑎^𝑥)
- Logarithmic function (e.g., ℎ(𝑥) = 𝑙𝑜𝑔𝑎 𝑥).
Class Exercise Highlights
- Identify polynomial function characteristics, such as degree, leading coefficient, and constant.
- Example: For 𝑦 = −5𝑥^4 + 𝑥^3 + 1, the Degree = 4, Leading Coefficient = -5, Constant = 1.
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