Podcast
Questions and Answers
What is the definition of a function?
What is the definition of a function?
- A relationship that must involve only linear equations.
- A mapping rule that uniquely assigns one output to each input from the domain. (correct)
- A random association between inputs and outputs.
- A mapping that assigns multiple outputs for a single input.
Which of the following best describes the domain of a function?
Which of the following best describes the domain of a function?
- The count of elements in the function's co-domain.
- The set of elements that may be inserted into the function. (correct)
- The range of values that the function outputs.
- The output values produced by the function.
What role do logarithmic functions play in relation to exponential functions?
What role do logarithmic functions play in relation to exponential functions?
- They cannot be used with exponential values.
- They are the inverse functions of exponential functions. (correct)
- They serve as the coefficients of exponential functions.
- They are the same type of function but with different representations.
How is a relation established with the temperature data measured by Lisa?
How is a relation established with the temperature data measured by Lisa?
Which of the following statements about functions is TRUE?
Which of the following statements about functions is TRUE?
What are trigonometric functions primarily used for?
What are trigonometric functions primarily used for?
What is a necessary property for a function to be invertible?
What is a necessary property for a function to be invertible?
What is an example of a mapping rule dependent on conditions?
What is an example of a mapping rule dependent on conditions?
What does the degree of a polynomial function indicate?
What does the degree of a polynomial function indicate?
Which family of functions is used to represent growth and decay processes?
Which family of functions is used to represent growth and decay processes?
What is true about the inverse function of an exponential function?
What is true about the inverse function of an exponential function?
What characteristic of trigonometric functions makes them suitable for modeling periodic relationships?
What characteristic of trigonometric functions makes them suitable for modeling periodic relationships?
Which is the base of the natural exponential function?
Which is the base of the natural exponential function?
What determines whether a graph consists of points or a continuous line?
What determines whether a graph consists of points or a continuous line?
In a linear function, what does the parameter 'a' represent?
In a linear function, what does the parameter 'a' represent?
Which of the following is true about the identity function id(x): ℝ → ℝ?
Which of the following is true about the identity function id(x): ℝ → ℝ?
What is necessary for two functions to be considered identical?
What is necessary for two functions to be considered identical?
If a linear function has a slope 'a' that is less than 0, what direction does the graph run?
If a linear function has a slope 'a' that is less than 0, what direction does the graph run?
What factors influence the shape of a quadratic function?
What factors influence the shape of a quadratic function?
What is the standard form of a quadratic function?
What is the standard form of a quadratic function?
When is a function considered a constant function?
When is a function considered a constant function?
Which type of function can be composed of two linear functions?
Which type of function can be composed of two linear functions?
What happens if the parameter 'a' in a quadratic function is equal to 0?
What happens if the parameter 'a' in a quadratic function is equal to 0?
What is the expected appearance of a graph representing real numbers or a subset thereof?
What is the expected appearance of a graph representing real numbers or a subset thereof?
What is the special form of a quadratic function when a = 1, b = 0, and c = 0?
What is the special form of a quadratic function when a = 1, b = 0, and c = 0?
For natural numbers, what is the smallest value included in the domain when defining a constant function?
For natural numbers, what is the smallest value included in the domain when defining a constant function?
What is required for a function to have an inverse?
What is required for a function to have an inverse?
Which of the following functions is not invertible?
Which of the following functions is not invertible?
For the function g(x) = 1/3^x, which statement is true about its behavior?
For the function g(x) = 1/3^x, which statement is true about its behavior?
Which composition confirms that two functions are inverses of each other?
Which composition confirms that two functions are inverses of each other?
Which of the following represents the correct inverse of the function f(x) = x^2 defined on ℝ+?
Which of the following represents the correct inverse of the function f(x) = x^2 defined on ℝ+?
What does the term bijective signify in the context of functions?
What does the term bijective signify in the context of functions?
Why is the function h(x) = -2x + 3 invertible?
Why is the function h(x) = -2x + 3 invertible?
How can the functions h and k be graphically distinguished regarding their inverses?
How can the functions h and k be graphically distinguished regarding their inverses?
What happens when a function is not injective in relation to its inverse?
What happens when a function is not injective in relation to its inverse?
Which of the following describes the composition of functions when checking for inverses?
Which of the following describes the composition of functions when checking for inverses?
What characterizes a strictly monotonically increasing function?
What characterizes a strictly monotonically increasing function?
What is the effect of a function having a base of 1 in an exponential function?
What is the effect of a function having a base of 1 in an exponential function?
If a function is bijective, what implication does this have for its inverse?
If a function is bijective, what implication does this have for its inverse?
Which statement holds true for the function k(x) = -1/2x + 32?
Which statement holds true for the function k(x) = -1/2x + 32?
What is the relationship between the argument and the function value in a defined function?
What is the relationship between the argument and the function value in a defined function?
Which of the following describes a constant function?
Which of the following describes a constant function?
What is the identity function defined as?
What is the identity function defined as?
In the context of functions, what does the term 'co-domain' refer to?
In the context of functions, what does the term 'co-domain' refer to?
Which of the following statements is true about a function defined as f: A → B?
Which of the following statements is true about a function defined as f: A → B?
What does the absolute value function do with negative input values?
What does the absolute value function do with negative input values?
If a mapping shows that f(3) = c and f(4) = c, what type of function is this likely to be?
If a mapping shows that f(3) = c and f(4) = c, what type of function is this likely to be?
What is true about the function set f: A → B if A and B are the same set?
What is true about the function set f: A → B if A and B are the same set?
What characterizes a surjective function?
What characterizes a surjective function?
Which of the following describes how the graph of a function is represented?
Which of the following describes how the graph of a function is represented?
In a situation where multiple arguments produce the same output, what is the implication for the function's properties?
In a situation where multiple arguments produce the same output, what is the implication for the function's properties?
Which of the following functions is an example of a surjective function given certain restrictions?
Which of the following functions is an example of a surjective function given certain restrictions?
In the case of a function from a set A to a set B, what must be true about the mapping?
In the case of a function from a set A to a set B, what must be true about the mapping?
Which statement correctly defines an injective function?
Which statement correctly defines an injective function?
Why is the function f: ℝ → ℝ, f(x) = x^2 not injective?
Why is the function f: ℝ → ℝ, f(x) = x^2 not injective?
What does the term 'image of a function' refer to?
What does the term 'image of a function' refer to?
What happens to the function values of g(x) = 1/(3^x) as x approaches infinity?
What happens to the function values of g(x) = 1/(3^x) as x approaches infinity?
In what scenario can a quadratic function be injective?
In what scenario can a quadratic function be injective?
What can be deduced if a function has elements in the codomain to which no arguments are mapped?
What can be deduced if a function has elements in the codomain to which no arguments are mapped?
For the function f(x) = 3^x, what characterizes its behavior as x approaches negative infinity?
For the function f(x) = 3^x, what characterizes its behavior as x approaches negative infinity?
What is a bijective function?
What is a bijective function?
In what type of function are all arguments mapped to their respective negative outputs?
In what type of function are all arguments mapped to their respective negative outputs?
Which equation represents the bacterial growth after t hours if the initial area is 80 mm² and it increases by 25% every hour?
Which equation represents the bacterial growth after t hours if the initial area is 80 mm² and it increases by 25% every hour?
In the exponential function for air pressure, what value is used for the base a when the air pressure halves every 5.5 km?
In the exponential function for air pressure, what value is used for the base a when the air pressure halves every 5.5 km?
An example of a linear function that is not bijective is?
An example of a linear function that is not bijective is?
Why is the function of Lisa’s temperature data not surjective?
Why is the function of Lisa’s temperature data not surjective?
Which of the following correctly defines the natural exponential function?
Which of the following correctly defines the natural exponential function?
What happens to the amount in a bank account under continuous compounding as the compounding periods per year increase?
What happens to the amount in a bank account under continuous compounding as the compounding periods per year increase?
Which function would be considered invertible?
Which function would be considered invertible?
What does the composition of two functions that results in the identity function imply?
What does the composition of two functions that results in the identity function imply?
What is the inverse function of an exponential function y = a^x with respect to its domain?
What is the inverse function of an exponential function y = a^x with respect to its domain?
If a function f: A → B is not injective, which of the following is necessarily true?
If a function f: A → B is not injective, which of the following is necessarily true?
If the altitude increases by 5.5 km, what happens to the air pressure according to the given description?
If the altitude increases by 5.5 km, what happens to the air pressure according to the given description?
What form does the decay process of air pressure take based on the initial value and the exponential decay factor?
What form does the decay process of air pressure take based on the initial value and the exponential decay factor?
Which feature defines a function that cannot be invertible?
Which feature defines a function that cannot be invertible?
What must hold true for a function f: A → B and its inverse f^{-1}: B → A?
What must hold true for a function f: A → B and its inverse f^{-1}: B → A?
Which of the following describes the relationship of the limit that determines Euler's constant?
Which of the following describes the relationship of the limit that determines Euler's constant?
What does the general formula for compound interest include when interest is compounded n times per year?
What does the general formula for compound interest include when interest is compounded n times per year?
What is the co-domain of the function f: ℝ → ℝ+ extbackslash{0} described in the content?
What is the co-domain of the function f: ℝ → ℝ+ extbackslash{0} described in the content?
In terms of exponential functions, bases greater than 1 result in what type of growth behavior?
In terms of exponential functions, bases greater than 1 result in what type of growth behavior?
What effect does the constant c ≠ 0 have on a quadratic function?
What effect does the constant c ≠ 0 have on a quadratic function?
What happens to the vertex of a quadratic function when the coefficient b ≠ 0 is introduced?
What happens to the vertex of a quadratic function when the coefficient b ≠ 0 is introduced?
Which statement about higher-order polynomial functions is true?
Which statement about higher-order polynomial functions is true?
What is the correct general form of a third-order polynomial function?
What is the correct general form of a third-order polynomial function?
What defines a surjective function?
What defines a surjective function?
How does the composition of functions operate?
How does the composition of functions operate?
What characterizes the composition of two functions in terms of commutativity?
What characterizes the composition of two functions in terms of commutativity?
Given the functions f(x) = 2x - 1 and g(x) = x^2, what is the composition g(f(x))?
Given the functions f(x) = 2x - 1 and g(x) = x^2, what is the composition g(f(x))?
Which of the following describes a quadratic function?
Which of the following describes a quadratic function?
Which of the following statements is incorrect regarding cubic functions?
Which of the following statements is incorrect regarding cubic functions?
How does a negative coefficient for a quadratic function's leading term affect its graph?
How does a negative coefficient for a quadratic function's leading term affect its graph?
What is the effect of higher degree polynomial functions compared to quadratic functions?
What is the effect of higher degree polynomial functions compared to quadratic functions?
What does the notation f: A -> B signify in function terminology?
What does the notation f: A -> B signify in function terminology?
Which equation represents an upward-opening parabola?
Which equation represents an upward-opening parabola?
What is the inverse function of the exponential function represented by g(x) = log_a(x)?
What is the inverse function of the exponential function represented by g(x) = log_a(x)?
In the context of the natural exponential function, what does the term 'continuous growth rate' refer to?
In the context of the natural exponential function, what does the term 'continuous growth rate' refer to?
Which of the following equations is correctly used to find the time for the bacterial area to double?
Which of the following equations is correctly used to find the time for the bacterial area to double?
What do the functions sin(x) and cos(x) represent on the unit circle?
What do the functions sin(x) and cos(x) represent on the unit circle?
Which of the following identities is true for all x in relation to the sine and cosine functions?
Which of the following identities is true for all x in relation to the sine and cosine functions?
How does one express an exponential function of the form f(x) = f_0 · a^x using the natural exponential function?
How does one express an exponential function of the form f(x) = f_0 · a^x using the natural exponential function?
For any angle x in radians, what is true about the sine and cosine values?
For any angle x in radians, what is true about the sine and cosine values?
Which transformation does the exponential function undergo when expressed in logarithmic form?
Which transformation does the exponential function undergo when expressed in logarithmic form?
What is the connection between the graphs of the functions a^x and log_a(x)?
What is the connection between the graphs of the functions a^x and log_a(x)?
In the unit circle, what does the angle α represent?
In the unit circle, what does the angle α represent?
What is the value of cos(3π/2)?
What is the value of cos(3π/2)?
What are the values of sin(π) and cos(π)?
What are the values of sin(π) and cos(π)?
What happens to the sine and cosine functions when the angle x increases by 2π?
What happens to the sine and cosine functions when the angle x increases by 2π?
Why can't Lisa's temperature data be described with an exponential or logarithmic function?
Why can't Lisa's temperature data be described with an exponential or logarithmic function?
What is the periodicity of the sine and cosine functions?
What is the periodicity of the sine and cosine functions?
What is the image of the tangent function?
What is the image of the tangent function?
In which interval is the sine function strictly monotonically increasing?
In which interval is the sine function strictly monotonically increasing?
What is the formula to calculate the amplitude of an oscillating function?
What is the formula to calculate the amplitude of an oscillating function?
What is the domain restriction for the cotangent function?
What is the domain restriction for the cotangent function?
What describes the average level of the oscillation in the function model?
What describes the average level of the oscillation in the function model?
What is the inverse function of the sine function on the restricted interval?
What is the inverse function of the sine function on the restricted interval?
Which of the following trigonometric functions is strictly monotonically decreasing on its restricted interval?
Which of the following trigonometric functions is strictly monotonically decreasing on its restricted interval?
What is the formula for the period parameter $b$ in the oscillating function?
What is the formula for the period parameter $b$ in the oscillating function?
Which parameter represents the upward and downward deviations in the oscillation model?
Which parameter represents the upward and downward deviations in the oscillation model?
What values are excluded from the domain of the tangent function?
What values are excluded from the domain of the tangent function?
What describes the relationship between sin and cos functions based on their graphs?
What describes the relationship between sin and cos functions based on their graphs?
What defines the cotangent in terms of sine and cosine?
What defines the cotangent in terms of sine and cosine?
Flashcards
Function
Function
A function is a mapping rule that uniquely assigns an element from a set (called the codomain) to each element from another set (called the domain).
Domain of a function
Domain of a function
The set of all possible input values (elements) for a function.
Codomain of a function
Codomain of a function
The set of all possible output values (elements) for a function.
Mapping rule
Mapping rule
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Relation
Relation
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Attribute
Attribute
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Elementary function
Elementary function
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Inverse function
Inverse function
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Function Graph
Function Graph
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Identity Function
Identity Function
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Constant Function
Constant Function
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Absolute Value Function
Absolute Value Function
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Linear Function
Linear Function
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Slope of a linear function
Slope of a linear function
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Y-intercept of a linear function
Y-intercept of a linear function
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Quadratic Function
Quadratic Function
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Normal Parabola
Normal Parabola
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Composition of Functions
Composition of Functions
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Real Data in Functions
Real Data in Functions
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Temperature Curve
Temperature Curve
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Function Definition
Function Definition
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Domain vs. Codomain
Domain vs. Codomain
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Function Notation
Function Notation
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Argument / Input Value
Argument / Input Value
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Function Value / Output
Function Value / Output
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Image of a Function
Image of a Function
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Uniqueness of Function Values
Uniqueness of Function Values
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Graph of a Function
Graph of a Function
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Domain of a function - Example 1
Domain of a function - Example 1
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Codomain of a function - Example 1
Codomain of a function - Example 1
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Function Value - Example 1
Function Value - Example 1
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Function Value - Example 2
Function Value - Example 2
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Stretched Parabola
Stretched Parabola
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Compressed Parabola
Compressed Parabola
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Parabola Opening Downwards
Parabola Opening Downwards
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Vertical Shift of a Parabola
Vertical Shift of a Parabola
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Horizontal Shift of a Parabola
Horizontal Shift of a Parabola
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Third-Order Polynomial Function
Third-Order Polynomial Function
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Polynomial Function of Degree n
Polynomial Function of Degree n
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Commutative Property in Composition
Commutative Property in Composition
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Surjective Function
Surjective Function
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Valley or Mountain of a Quadratic Function
Valley or Mountain of a Quadratic Function
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Modeling Lisa's Temperature Data
Modeling Lisa's Temperature Data
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Suitable Function for Lisa's Data
Suitable Function for Lisa's Data
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Polynomial Function for Temperature Data
Polynomial Function for Temperature Data
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Limitations of Polynomial Functions for Temperature Data
Limitations of Polynomial Functions for Temperature Data
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Invertible Function
Invertible Function
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Invertible Function (Another perspective)
Invertible Function (Another perspective)
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Example of a Non-Surjective Function
Example of a Non-Surjective Function
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Example of a Non-Injective Function
Example of a Non-Injective Function
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Temperature Curve in Functions
Temperature Curve in Functions
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What does surjective mean in simple terms?
What does surjective mean in simple terms?
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What does injective mean in simple terms?
What does injective mean in simple terms?
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Polynomial Function
Polynomial Function
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Exponential Function
Exponential Function
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Natural Exponential Function
Natural Exponential Function
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Logarithmic Function
Logarithmic Function
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Trigonometric Functions
Trigonometric Functions
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Identity Function (id)
Identity Function (id)
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General Exponential Function
General Exponential Function
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Base of an Exponential Function
Base of an Exponential Function
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Exponent of an Exponential Function
Exponent of an Exponential Function
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Strictly Monotonically Increasing Function
Strictly Monotonically Increasing Function
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Strictly Monotonically Decreasing Function
Strictly Monotonically Decreasing Function
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Monotonicity of Functions
Monotonicity of Functions
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Injective Function (One-to-one)
Injective Function (One-to-one)
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Surjective Function (Onto)
Surjective Function (Onto)
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Restriction of a Function
Restriction of a Function
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Asymptote
Asymptote
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x → ∞
x → ∞
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x → –∞
x → –∞
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Euler's Constant (e)
Euler's Constant (e)
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Compound Interest
Compound Interest
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Continuous Compounding
Continuous Compounding
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Base of a Logarithm
Base of a Logarithm
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Cosine Function
Cosine Function
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Periodicity
Periodicity
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Tangent Function
Tangent Function
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Cotangent Function
Cotangent Function
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Domain of a Trigonometric Function
Domain of a Trigonometric Function
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Inverse Trigonometric Function
Inverse Trigonometric Function
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Arcsine Function
Arcsine Function
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Arccosine Function
Arccosine Function
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Arctangent Function
Arctangent Function
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Arccotangent Function
Arccotangent Function
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Amplitude of an Oscillation
Amplitude of an Oscillation
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Average Level in Oscillations
Average Level in Oscillations
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Modeling Temperature Data
Modeling Temperature Data
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Sine Function for Temperature Modeling
Sine Function for Temperature Modeling
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Natural Logarithm
Natural Logarithm
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Exponential Growth Rate
Exponential Growth Rate
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Solving Exponential Equations
Solving Exponential Equations
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Unit Circle
Unit Circle
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Cosine (cos α)
Cosine (cos α)
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Sine (sin α)
Sine (sin α)
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Radian Measure
Radian Measure
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Trigonometric Functions (sin, cos, tan, cot)
Trigonometric Functions (sin, cos, tan, cot)
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Periodicity of Sine and Cosine
Periodicity of Sine and Cosine
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Pythagorean Identity
Pythagorean Identity
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Trigonometric Function Values
Trigonometric Function Values
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Monotonic Functions
Monotonic Functions
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Growth or Decay Processes
Growth or Decay Processes
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Exponential and Logarithmic Functions for Data
Exponential and Logarithmic Functions for Data
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Study Notes
Functions and Their Properties
- A function (or map) assigns one element from a set B (co-domain) to each element from a set A (domain).
- Both domain and co-domain are non-empty.
- Notation: f: A → B (f is a function from A to B)
- Argument (input) x ∈ A is inserted into the function f.
- Function value f(x) ∈ B is the element from the co-domain to which x is mapped.
- Image of f: Im(f) - Set of all function values.
- A function value is unique for a given input. Several inputs can map to the same output value.
Types of Functions
- Identity function: Maps each element to itself (id(x) = x).
- Constant function: Assigns a single element to all inputs (const(x) = c).
- Absolute value function: Maps negative inputs to positive outputs. |x| = x if x ≥ 0, -x if x < 0
Graphical Representation
- Graph of a function: Visual representation in a coordinate system.
- Point pairs (x, f(x)) represent the graph.
Elementary Functions
-
Linear function: f(x) = ax + b (a, b are real numbers). The graph is a straight line.
- 'a' determines the slope of the line.
- 'b' is the y-intercept.
-
Quadratic function: f(x) = ax² + bx + c (a, b, c are real numbers, a ≠ 0). The graph is a parabola.
- Parabola opens upwards if a > 0.
- Parabola opens downwards if a < 0.
- Parameter 'c' shifts the parabola vertically.
- Parameter 'b' shifts the vertex horizontally.
-
Third-order polynomial function: f(x) = ax³ + bx² + cx + d
-
Polynomial functions of degree n: A generalized polynomial function, expressed using the sum symbol Σ, describing terms with increasing input variable powers.
Composition of Functions
- Composition: Combining functions.
- Given functions f: A → B and g C → D, where all values in f(A) are also in the domain of g (f(x) ∈ C for all x ∈ A), the composition g ∘ f: A → D has the form g(f(x)).
- Calculation order: Inner function first, then outer function.
- Composition is not commutative (f ∘ g ≠ g ∘ f).
Properties of Functions
-
Surjective: Function f: A → B is surjective if every element in the co-domain B has at least one corresponding element in the domain A.
-
Injective: Function f is injective if different input values result in different output values. That is, if f(x₁) = f(x₂), then x₁ = x₂.
-
Bijective: Function f is bijective if it is both surjective and injective.
Invertible Functions
- Invertible function: A function that has an inverse function.
- Inverse function (f⁻¹): A function g that reverses the effect of function f.
- g ∘ f = identity and f ∘ g = identity
- For a function to be invertible, it must be bijective.
Exponential Functions
- General exponential function: f(x) = aˣ, where a is a positive constant and x is any number.
- Used for growth or decay models.
Logarithmic Functions
- Logarithmic function: g(x) = logₐx, the inverse of exponential function.
- loga(aˣ) = x and a logₐx = x
Natural Exponential Function
- Natural exponential function: exp(x) = eˣ.
- Exp(x) is used for modeling growth and decay processes and continuously-compounded interest calculations.
- Euler's constant (e) is ≈ 2.718.
Trigonometric Functions
- Defined using the unit circle.
- Includes sine, cosine, tangent, and cotangent.
- Periodic functions (values repeat).
- Inverse trigonometric functions exist when the domain of the original function is restricted to an interval where the function is strictly monotonically increasing or decreasing. These restricted functions are thus bijective and invertible.
- Includes arcsine, arccosine, arctangent, and arccotangent.
Studying That Suits You
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Description
This quiz covers the fundamental concepts of functions, including their definitions, types, and graphical representations. Explore the different types of functions, such as identity, constant, and absolute value functions, and understand how they map elements from the domain to the co-domain. Test your knowledge on these essential mathematical concepts.