Functions and Their Properties
121 Questions
12 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

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?

  • 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?

  • 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?

<p>By assigning one specific temperature to each unique time entry. (B)</p> Signup and view all the answers

Which of the following statements about functions is TRUE?

<p>Functions represent relationships between quantities in distinct ways. (B)</p> Signup and view all the answers

What are trigonometric functions primarily used for?

<p>To represent circular relationships and periodic phenomena. (B)</p> Signup and view all the answers

What is a necessary property for a function to be invertible?

<p>It must have unique outputs for each input. (A)</p> Signup and view all the answers

What is an example of a mapping rule dependent on conditions?

<p>Mapping temperatures only when the day is sunny. (A)</p> Signup and view all the answers

What does the degree of a polynomial function indicate?

<p>The highest power of the input variable (B)</p> Signup and view all the answers

Which family of functions is used to represent growth and decay processes?

<p>Exponential functions (A)</p> Signup and view all the answers

What is true about the inverse function of an exponential function?

<p>It is unique and invertible. (D)</p> Signup and view all the answers

What characteristic of trigonometric functions makes them suitable for modeling periodic relationships?

<p>They exhibit periodicity in their values. (D)</p> Signup and view all the answers

Which is the base of the natural exponential function?

<p>e (A)</p> Signup and view all the answers

What determines whether a graph consists of points or a continuous line?

<p>The domain and co-domain of the function. (D)</p> Signup and view all the answers

In a linear function, what does the parameter 'a' represent?

<p>The slope of the line. (B)</p> Signup and view all the answers

Which of the following is true about the identity function id(x): ℝ → ℝ?

<p>It is represented by the function id(x) = x. (D)</p> Signup and view all the answers

What is necessary for two functions to be considered identical?

<p>Both A and B must match. (B)</p> Signup and view all the answers

If a linear function has a slope 'a' that is less than 0, what direction does the graph run?

<p>From the top left to the bottom right. (B)</p> Signup and view all the answers

What factors influence the shape of a quadratic function?

<p>The values of a, b, and c in the function. (B)</p> Signup and view all the answers

What is the standard form of a quadratic function?

<p>f(x) = a ⋅ x^2 + b ⋅ x + c. (D)</p> Signup and view all the answers

When is a function considered a constant function?

<p>When the parameter a equals 0. (D)</p> Signup and view all the answers

Which type of function can be composed of two linear functions?

<p>Absolute value function. (A)</p> Signup and view all the answers

What happens if the parameter 'a' in a quadratic function is equal to 0?

<p>The function transforms into a linear function. (B)</p> Signup and view all the answers

What is the expected appearance of a graph representing real numbers or a subset thereof?

<p>A continuous curve. (D)</p> Signup and view all the answers

What is the special form of a quadratic function when a = 1, b = 0, and c = 0?

<p>A normal parabola. (A)</p> Signup and view all the answers

For natural numbers, what is the smallest value included in the domain when defining a constant function?

<ol> <li>(C)</li> </ol> Signup and view all the answers

What is required for a function to have an inverse?

<p>The function must be bijective. (A)</p> Signup and view all the answers

Which of the following functions is not invertible?

<p>f(x) = x^2 (A)</p> Signup and view all the answers

For the function g(x) = 1/3^x, which statement is true about its behavior?

<p>It is strictly monotonically decreasing. (B)</p> Signup and view all the answers

Which composition confirms that two functions are inverses of each other?

<p>f ∘ g = id (A)</p> Signup and view all the answers

Which of the following represents the correct inverse of the function f(x) = x^2 defined on ℝ+?

<p>g(x) = √x (A)</p> Signup and view all the answers

What does the term bijective signify in the context of functions?

<p>All outputs are unique for their respective inputs. (D)</p> Signup and view all the answers

Why is the function h(x) = -2x + 3 invertible?

<p>It is linear and bijective. (D)</p> Signup and view all the answers

How can the functions h and k be graphically distinguished regarding their inverses?

<p>They reflect across the identity function. (B)</p> Signup and view all the answers

What happens when a function is not injective in relation to its inverse?

<p>It can have multiple outputs for one input. (D)</p> Signup and view all the answers

Which of the following describes the composition of functions when checking for inverses?

<p>It must return the original input without variations. (A)</p> Signup and view all the answers

What characterizes a strictly monotonically increasing function?

<p>The function values always increase with input increases. (B)</p> Signup and view all the answers

What is the effect of a function having a base of 1 in an exponential function?

<p>The function is constantly equal to 1. (B)</p> Signup and view all the answers

If a function is bijective, what implication does this have for its inverse?

<p>The inverse exists and is also bijective. (D)</p> Signup and view all the answers

Which statement holds true for the function k(x) = -1/2x + 32?

<p>It is a linear and invertible function. (A)</p> Signup and view all the answers

What is the relationship between the argument and the function value in a defined function?

<p>Each argument maps to exactly one function value. (C)</p> Signup and view all the answers

Which of the following describes a constant function?

<p>It maps all inputs to the same single element. (D)</p> Signup and view all the answers

What is the identity function defined as?

<p>f(x) = x for all x in the domain. (C)</p> Signup and view all the answers

In the context of functions, what does the term 'co-domain' refer to?

<p>The set of values that a function can take. (C)</p> Signup and view all the answers

Which of the following statements is true about a function defined as f: A → B?

<p>Inputs from A must map to unique outputs in B. (A)</p> Signup and view all the answers

What does the absolute value function do with negative input values?

<p>It converts them to positive values. (D)</p> Signup and view all the answers

If a mapping shows that f(3) = c and f(4) = c, what type of function is this likely to be?

<p>A constant function. (B)</p> Signup and view all the answers

What is true about the function set f: A → B if A and B are the same set?

<p>It may be an identity function. (C)</p> Signup and view all the answers

What characterizes a surjective function?

<p>For each element in the co-domain, there is at least one corresponding element in the domain. (D)</p> Signup and view all the answers

Which of the following describes how the graph of a function is represented?

<p>Both input values and function values are plotted as points. (B)</p> Signup and view all the answers

In a situation where multiple arguments produce the same output, what is the implication for the function's properties?

<p>The function remains valid as long as inputs remain unique. (A)</p> Signup and view all the answers

Which of the following functions is an example of a surjective function given certain restrictions?

<p>f(x) = x^2 with co-domain ℝ+ (A)</p> Signup and view all the answers

In the case of a function from a set A to a set B, what must be true about the mapping?

<p>Every element of A must map to a unique element of B. (C)</p> Signup and view all the answers

Which statement correctly defines an injective function?

<p>If two outputs are equal, the corresponding inputs must also be equal. (B)</p> Signup and view all the answers

Why is the function f: ℝ → ℝ, f(x) = x^2 not injective?

<p>Multiple input values can yield the same output. (D)</p> Signup and view all the answers

What does the term 'image of a function' refer to?

<p>The set of all possible function values produced. (B)</p> Signup and view all the answers

What happens to the function values of g(x) = 1/(3^x) as x approaches infinity?

<p>They approach 0. (C)</p> Signup and view all the answers

In what scenario can a quadratic function be injective?

<p>If the domain is restricted to positive numbers only. (A)</p> Signup and view all the answers

What can be deduced if a function has elements in the codomain to which no arguments are mapped?

<p>Some values in the co-domain are unused. (D)</p> Signup and view all the answers

For the function f(x) = 3^x, what characterizes its behavior as x approaches negative infinity?

<p>The function values approach 0. (B)</p> Signup and view all the answers

What is a bijective function?

<p>A function that is both surjective and injective. (D)</p> Signup and view all the answers

In what type of function are all arguments mapped to their respective negative outputs?

<p>There is no such function. (D)</p> Signup and view all the answers

Which equation represents the bacterial growth after t hours if the initial area is 80 mm² and it increases by 25% every hour?

<p>f(t) = 80 * 1.25^t (D)</p> Signup and view all the answers

In the exponential function for air pressure, what value is used for the base a when the air pressure halves every 5.5 km?

<p>0.8816 (B)</p> Signup and view all the answers

An example of a linear function that is not bijective is?

<p>f(x) = 3 (C)</p> Signup and view all the answers

Why is the function of Lisa’s temperature data not surjective?

<p>It does not cover all possible temperature values within its co-domain. (B)</p> Signup and view all the answers

Which of the following correctly defines the natural exponential function?

<p>f(x) = e^x where e = 2.71828 (B)</p> Signup and view all the answers

What happens to the amount in a bank account under continuous compounding as the compounding periods per year increase?

<p>It converges to an exponential growth formula. (B)</p> Signup and view all the answers

Which function would be considered invertible?

<p>A function that is both surjective and injective. (D)</p> Signup and view all the answers

What does the composition of two functions that results in the identity function imply?

<p>The functions are inverses of each other. (A)</p> Signup and view all the answers

What is the inverse function of an exponential function y = a^x with respect to its domain?

<p>x = log_a(y) (A)</p> Signup and view all the answers

If a function f: A → B is not injective, which of the following is necessarily true?

<p>Some inputs map to the same output. (D)</p> Signup and view all the answers

If the altitude increases by 5.5 km, what happens to the air pressure according to the given description?

<p>It halves. (B)</p> Signup and view all the answers

What form does the decay process of air pressure take based on the initial value and the exponential decay factor?

<p>p(x) = p0 * 0.8816^x (A)</p> Signup and view all the answers

Which feature defines a function that cannot be invertible?

<p>If it is surjective but not injective. (D)</p> Signup and view all the answers

What must hold true for a function f: A → B and its inverse f^{-1}: B → A?

<p>The compositions g ∘ f and f ∘ g must equal the identity function. (D)</p> Signup and view all the answers

Which of the following describes the relationship of the limit that determines Euler's constant?

<p>lim n→∞ (1 + 1/n)^n = e (D)</p> Signup and view all the answers

What does the general formula for compound interest include when interest is compounded n times per year?

<p>A = P(1 + r/n)^(nt) (B)</p> Signup and view all the answers

What is the co-domain of the function f: ℝ → ℝ+ extbackslash{0} described in the content?

<p>All positive real numbers excluding zero. (A)</p> Signup and view all the answers

In terms of exponential functions, bases greater than 1 result in what type of growth behavior?

<p>Exponential growth. (D)</p> Signup and view all the answers

What effect does the constant c ≠ 0 have on a quadratic function?

<p>It shifts the graph vertically by c units. (A)</p> Signup and view all the answers

What happens to the vertex of a quadratic function when the coefficient b ≠ 0 is introduced?

<p>The vertex is shifted left or right along the x-axis. (B)</p> Signup and view all the answers

Which statement about higher-order polynomial functions is true?

<p>They can have multiple valleys and mountains. (A)</p> Signup and view all the answers

What is the correct general form of a third-order polynomial function?

<p>f(x) = ax^3 + bx^2 + cx + d (C)</p> Signup and view all the answers

What defines a surjective function?

<p>At least one input value corresponds to each element in the co-domain. (C)</p> Signup and view all the answers

How does the composition of functions operate?

<p>The inner function is applied first. (D)</p> Signup and view all the answers

What characterizes the composition of two functions in terms of commutativity?

<p>The order of functions affects the result. (C)</p> Signup and view all the answers

Given the functions f(x) = 2x - 1 and g(x) = x^2, what is the composition g(f(x))?

<p>4x^2 - 4x + 1 (A)</p> Signup and view all the answers

Which of the following describes a quadratic function?

<p>It is the simplest form of a polynomial function. (A)</p> Signup and view all the answers

Which of the following statements is incorrect regarding cubic functions?

<p>They can never intersect the x-axis more than three times. (D)</p> Signup and view all the answers

How does a negative coefficient for a quadratic function's leading term affect its graph?

<p>The graph opens downwards. (C)</p> Signup and view all the answers

What is the effect of higher degree polynomial functions compared to quadratic functions?

<p>Higher degree polynomials can have more complex shapes. (C)</p> Signup and view all the answers

What does the notation f: A -> B signify in function terminology?

<p>f is a function that relates elements from set A to set B. (A)</p> Signup and view all the answers

Which equation represents an upward-opening parabola?

<p>f(x) = 0.5x^2 - 4 (A)</p> Signup and view all the answers

What is the inverse function of the exponential function represented by g(x) = log_a(x)?

<p>g(x) = ln(x) (B)</p> Signup and view all the answers

In the context of the natural exponential function, what does the term 'continuous growth rate' refer to?

<p>The constant r associated with ln(a) (C)</p> Signup and view all the answers

Which of the following equations is correctly used to find the time for the bacterial area to double?

<p>160 = 80 · 1.25^x (C)</p> Signup and view all the answers

What do the functions sin(x) and cos(x) represent on the unit circle?

<p>The projections of point P onto the x-axis and y-axis, respectively (B)</p> Signup and view all the answers

Which of the following identities is true for all x in relation to the sine and cosine functions?

<p>sin(x + π) = -sin x (B)</p> Signup and view all the answers

How does one express an exponential function of the form f(x) = f_0 · a^x using the natural exponential function?

<p>f(x) = e^(r · x) where r = ln(a) (B)</p> Signup and view all the answers

For any angle x in radians, what is true about the sine and cosine values?

<p>Both sin x and cos x are bounded within [-1, 1] (C)</p> Signup and view all the answers

Which transformation does the exponential function undergo when expressed in logarithmic form?

<p>The base remains the same (D)</p> Signup and view all the answers

What is the connection between the graphs of the functions a^x and log_a(x)?

<p>They are inverses and mirror images across the identity line (A)</p> Signup and view all the answers

In the unit circle, what does the angle α represent?

<p>The length of the arc on the unit circle (A)</p> Signup and view all the answers

What is the value of cos(3π/2)?

<p>0 (B)</p> Signup and view all the answers

What are the values of sin(π) and cos(π)?

<p>sin(π) = 0, cos(π) = -1 (D)</p> Signup and view all the answers

What happens to the sine and cosine functions when the angle x increases by 2π?

<p>They reset to their original values (A)</p> Signup and view all the answers

Why can't Lisa's temperature data be described with an exponential or logarithmic function?

<p>It does not show a consistent growth or decay pattern (B)</p> Signup and view all the answers

What is the periodicity of the sine and cosine functions?

<p>$2 ext{π}$ (A)</p> Signup and view all the answers

What is the image of the tangent function?

<p>$ ext{ℝ}$ (C)</p> Signup and view all the answers

In which interval is the sine function strictly monotonically increasing?

<p>[- rac{ ext{π}}{2}, rac{ ext{π}}{2}] (C)</p> Signup and view all the answers

What is the formula to calculate the amplitude of an oscillating function?

<p>$a = rac{y_{ ext{max}} - y_{ ext{min}}}{2}$ (A)</p> Signup and view all the answers

What is the domain restriction for the cotangent function?

<p>$ ext{ℝ} ackslash k ext{π}, k ext{ is an integer}$ (B)</p> Signup and view all the answers

What describes the average level of the oscillation in the function model?

<p>Average value $d$ (C)</p> Signup and view all the answers

What is the inverse function of the sine function on the restricted interval?

<p>Arcsine (C)</p> Signup and view all the answers

Which of the following trigonometric functions is strictly monotonically decreasing on its restricted interval?

<p>Cosine (B)</p> Signup and view all the answers

What is the formula for the period parameter $b$ in the oscillating function?

<p>$b = rac{2 ext{π}}{p}$ (B)</p> Signup and view all the answers

Which parameter represents the upward and downward deviations in the oscillation model?

<p>Parameter $a$ (D)</p> Signup and view all the answers

What values are excluded from the domain of the tangent function?

<p>$ rac{ ext{π}}{2} + k ext{π}, k ext{ is an integer}$ (D)</p> Signup and view all the answers

What describes the relationship between sin and cos functions based on their graphs?

<p>Cosine is a phase shift of sine (A)</p> Signup and view all the answers

What defines the cotangent in terms of sine and cosine?

<p>$ ext{cot} ext{x} = rac{ ext{cos} x}{ ext{sin} x}$ (B)</p> Signup and view all the answers

Flashcards

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

The set of all possible input values (elements) for a function.

Codomain of a function

The set of all possible output values (elements) for a function.

Mapping rule

The specific rule used by a function to determine the output for each input value.

Signup and view all the flashcards

Relation

A general association between variables.

Signup and view all the flashcards

Attribute

A property or characteristic of something.

Signup and view all the flashcards

Elementary function

A group of basic functions like polynomials, exponentials, logarithmic, and trigonometric functions.

Signup and view all the flashcards

Inverse function

A function that reverses the action of another function.

Signup and view all the flashcards

Function Graph

A visual representation of a function showing the relationship between input values (x) and output values (f(x)) as points plotted on a coordinate plane.

Signup and view all the flashcards

Identity Function

A function that maps each input value to itself. For example, id(x) = x.

Signup and view all the flashcards

Constant Function

A function that always returns the same output value, regardless of the input. For example, const(x) = 2.

Signup and view all the flashcards

Absolute Value Function

A function that outputs the positive value of its input. For example, |x| = x if x ≥ 0, and |x| = -x if x < 0.

Signup and view all the flashcards

Linear Function

A function with the structure f(x) = ax + b, where a and b are constants. Its graph is a straight line.

Signup and view all the flashcards

Slope of a linear function

The coefficient 'a' in the equation f(x) = ax + b, which determines how steep the line is.

Signup and view all the flashcards

Y-intercept of a linear function

The constant term 'b' in the equation f(x) = ax + b, where the line crosses the y-axis.

Signup and view all the flashcards

Quadratic Function

A function with the structure f(x) = ax² + bx + c, where a, b, and c are constants. Its graph is a parabola.

Signup and view all the flashcards

Normal Parabola

The simplest quadratic function, f(x) = x², where a = 1, b = 0, and c = 0.

Signup and view all the flashcards

Composition of Functions

Combining two or more functions to create a new function, where the output of one function becomes the input of the next.

Signup and view all the flashcards

Real Data in Functions

When working with real-world data, the domain might not be a set of real numbers, but rather discrete points or intervals.

Signup and view all the flashcards

Temperature Curve

A graphical representation of temperature values over a period of time, often shown as a line graph.

Signup and view all the flashcards

Function Definition

A function is a mapping rule that uniquely assigns an element from a set (the co-domain) to each element from another set (the domain).

Signup and view all the flashcards

Domain vs. Codomain

The domain is the set of all possible input values for a function, while the codomain is the set of all possible output values.

Signup and view all the flashcards

Function Notation

We write 'f: A → B' to indicate that 'f' is a function from set 'A' (the domain) to set 'B' (the co-domain).

Signup and view all the flashcards

Argument / Input Value

An element from the domain that is inserted into a function is called the argument or input value.

Signup and view all the flashcards

Function Value / Output

The result of applying a function to an input value is called the function value, which is an element from the codomain.

Signup and view all the flashcards

Image of a Function

The set of all function values for a function is called the image of the function, often written as Im(f).

Signup and view all the flashcards

Uniqueness of Function Values

A function must assign a unique function value for each input value. Different inputs can have the same output, but the same input can't have multiple outputs.

Signup and view all the flashcards

Graph of a Function

A visual representation of a function in a coordinate system. Points on the graph represent input-output pairs (x, f(x)).

Signup and view all the flashcards

Domain of a function - Example 1

Consider the set A = {1, 2, 3, 4}. This set is the possible input values for a function.

Signup and view all the flashcards

Codomain of a function - Example 1

Consider the set B = {a, b, c, d }. This set is the set of all possible output values for a function.

Signup and view all the flashcards

Function Value - Example 1

If the mapping rule for a function is f(1) = a, f(2) = b, f(3) = c, f(4) = c, then the function values corresponding to the inputs 1, 2, 3, and 4 are 'a', 'b', 'c', and 'c', respectively.

Signup and view all the flashcards

Function Value - Example 2

If the mapping rule for a function is f(x) = x^2, and we input x = 3, then the function value is f(3) = 3^2 = 9.

Signup and view all the flashcards

Stretched Parabola

A parabola with a shape that is wider or narrower than a standard parabola, determined by the value of 'a' in the equation.

Signup and view all the flashcards

Compressed Parabola

A parabola with a shape that is narrower than a standard parabola, due to the value of 'a' being between 0 and 1 in the equation.

Signup and view all the flashcards

Parabola Opening Downwards

A parabola whose shape opens downwards, caused by a negative value of 'a' in the equation.

Signup and view all the flashcards

Vertical Shift of a Parabola

Moving the parabola up or down along the y-axis by adding a constant 'c' to the equation.

Signup and view all the flashcards

Horizontal Shift of a Parabola

Moving the parabola left or right along the x-axis, determined by the term 'b' in the equation.

Signup and view all the flashcards

Third-Order Polynomial Function

A function where the highest power of the variable is 3, represented by the general form: f(x) = ax³ + bx² + cx + d.

Signup and view all the flashcards

Polynomial Function of Degree n

A function where the highest power of the variable is 'n', represented by the general form: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀.

Signup and view all the flashcards

Commutative Property in Composition

The order of functions in a composition matters, changing the order can change the result.

Signup and view all the flashcards

Surjective Function

A function where every element in the codomain has at least one corresponding input from the domain.

Signup and view all the flashcards

Valley or Mountain of a Quadratic Function

The turning point of a parabola, either a minimum (valley) or maximum (mountain) point, depending on the value of 'a'.

Signup and view all the flashcards

Modeling Lisa's Temperature Data

Using mathematical functions to represent and analyze the changes in Lisa's temperature over time.

Signup and view all the flashcards

Suitable Function for Lisa's Data

A function that can accurately represent the ups and downs of Lisa's temperature over time.

Signup and view all the flashcards

Polynomial Function for Temperature Data

A polynomial function may be suitable for Lisa's temperature data if the data shows multiple ups and downs.

Signup and view all the flashcards

Limitations of Polynomial Functions for Temperature Data

Even high-degree polynomial functions may not accurately model Lisa's temperature data if the data shows non-polynomial behavior.

Signup and view all the flashcards

Invertible Function

A function that has an inverse function.

Signup and view all the flashcards

Invertible Function (Another perspective)

A function that has an inverse function, meaning there's another function that can reverse its actions.

Signup and view all the flashcards

Example of a Non-Surjective Function

A function where there exists at least one element in the codomain that has no corresponding input value in the domain.

Signup and view all the flashcards

Example of a Non-Injective Function

A function where at least two distinct input values are mapped to the same output value.

Signup and view all the flashcards

Temperature Curve in Functions

A graphical representation of temperature values over a period of time, often shown as a line graph.

Signup and view all the flashcards

What does surjective mean in simple terms?

A surjective function covers all possible outputs. It means that every possible output in the codomain has at least one corresponding input in the domain.

Signup and view all the flashcards

What does injective mean in simple terms?

An injective function maps distinct inputs to distinct outputs. This means that no two different inputs can produce the same output.

Signup and view all the flashcards

Polynomial Function

A function that combines terms with different powers of the input variable, where the highest power is the degree of the polynomial.

Signup and view all the flashcards

Exponential Function

A function where the input variable appears as an exponent, typically with a constant base.

Signup and view all the flashcards

Natural Exponential Function

A special exponential function with Euler's constant 'e' as the base.

Signup and view all the flashcards

Logarithmic Function

The inverse function of the exponential function, finding the exponent needed to reach a specific value.

Signup and view all the flashcards

Trigonometric Functions

Functions that relate angles to lengths of line segments in a unit circle, repeating in a cyclical pattern.

Signup and view all the flashcards

Identity Function (id)

A function that maps each input value to itself. For example, id(x) = x.

Signup and view all the flashcards

General Exponential Function

A function of the form f(x) = a^x, where 'a' is a constant positive base and 'x' is an exponent.

Signup and view all the flashcards

Base of an Exponential Function

The constant positive number 'a' in the exponential function f(x) = a^x. It determines the rate of growth or decay.

Signup and view all the flashcards

Exponent of an Exponential Function

The variable 'x' in the exponential function f(x) = a^x. It determines how many times the base is multiplied by itself.

Signup and view all the flashcards

Strictly Monotonically Increasing Function

A function where larger input values always lead to larger output values. For example, f(x) = 2x is strictly monotonically increasing.

Signup and view all the flashcards

Strictly Monotonically Decreasing Function

A function where larger input values always lead to smaller output values. For example, f(x) = -x is strictly monotonically decreasing.

Signup and view all the flashcards

Monotonicity of Functions

Describes how a function's output values change in relation to its input values. A function can be monotonically increasing, decreasing, or neither.

Signup and view all the flashcards

Injective Function (One-to-one)

A function where each output value is associated with a unique input value. No two different inputs map to the same output.

Signup and view all the flashcards

Surjective Function (Onto)

A function where every element in the codomain (set of possible outputs) is mapped to by at least one element in the domain (set of possible inputs).

Signup and view all the flashcards

Restriction of a Function

Limiting the domain and/or codomain of a function to create a new function. For example, restricting a function to only positive inputs or outputs.

Signup and view all the flashcards

Asymptote

A line that a function's graph approaches but never touches as the input values increase or decrease infinitely.

Signup and view all the flashcards

x → ∞

This means 'for x towards (plus) infinity'. It indicates that the input values are getting infinitely large in the positive direction.

Signup and view all the flashcards

x → –∞

This means 'for x towards minus infinity'. It indicates that the input values are getting infinitely large in the negative direction.

Signup and view all the flashcards

Euler's Constant (e)

A mathematical constant approximately equal to 2.71828. It serves as the base for the natural exponential function, and it appears in many areas of mathematics and science.

Signup and view all the flashcards

Compound Interest

Interest earned on both the principal amount and the accumulated interest from previous periods. It leads to faster growth compared to simple interest.

Signup and view all the flashcards

Continuous Compounding

Compounding interest infinitely many times per year, resulting in the fastest possible growth for a given interest rate.

Signup and view all the flashcards

Base of a Logarithm

The constant value that is raised to a power in a logarithm function. It's the starting point for the logarithmic calculations.

Signup and view all the flashcards

Cosine Function

A trigonometric function that relates an angle of a right triangle to the ratio of the adjacent side to the hypotenuse.

Signup and view all the flashcards

Periodicity

The property of a function where its values repeat at regular intervals.

Signup and view all the flashcards

Tangent Function

A trigonometric function defined as the ratio of the sine to the cosine: tan x = sin x / cos x.

Signup and view all the flashcards

Cotangent Function

A trigonometric function defined as the ratio of the cosine to the sine: cot x = cos x / sin x.

Signup and view all the flashcards

Domain of a Trigonometric Function

The set of all possible input values for which the function is defined.

Signup and view all the flashcards

Inverse Trigonometric Function

A function that 'undoes' a trigonometric function, giving back the angle.

Signup and view all the flashcards

Arcsine Function

The inverse function of the sine function, denoted arcsin(x) or sin⁻¹(x).

Signup and view all the flashcards

Arccosine Function

The inverse function of the cosine function, denoted arccos(x) or cos⁻¹(x).

Signup and view all the flashcards

Arctangent Function

The inverse function of the tangent function, denoted arctan(x) or tan⁻¹(x).

Signup and view all the flashcards

Arccotangent Function

The inverse function of the cotangent function, denoted arccot(x) or cot⁻¹(x).

Signup and view all the flashcards

Amplitude of an Oscillation

The maximum displacement from the equilibrium position of an oscillating function.

Signup and view all the flashcards

Average Level in Oscillations

The middle point around which an oscillating function fluctuates.

Signup and view all the flashcards

Modeling Temperature Data

Using mathematical functions to represent and analyze changes in temperature over time.

Signup and view all the flashcards

Sine Function for Temperature Modeling

A suitable function for modeling data that exhibits cyclical patterns, like temperature variations throughout a day.

Signup and view all the flashcards

Natural Logarithm

The inverse function of the natural exponential function (eˣ), denoted as ln(x).

Signup and view all the flashcards

Exponential Growth Rate

The constant 'r' that represents how much the function's value grows over a given time, expressed in the natural exponential form (eʳˣ).

Signup and view all the flashcards

Solving Exponential Equations

The process of finding the unknown value in an exponential equation by using logarithms.

Signup and view all the flashcards

Unit Circle

A circle with a radius of 1 and the center at the origin of a coordinate system.

Signup and view all the flashcards

Cosine (cos α)

The x-coordinate of a point on the unit circle, corresponding to the angle α.

Signup and view all the flashcards

Sine (sin α)

The y-coordinate of a point on the unit circle, corresponding to the angle α.

Signup and view all the flashcards

Radian Measure

A way of measuring angles using arc length on the unit circle.

Signup and view all the flashcards

Trigonometric Functions (sin, cos, tan, cot)

Functions that relate angles of a right triangle to the ratios of its sides.

Signup and view all the flashcards

Periodicity of Sine and Cosine

The property that the sine and cosine functions repeat their values after a certain interval (2π).

Signup and view all the flashcards

Pythagorean Identity

A fundamental trigonometric identity: sin²(x) + cos²(x) = 1.

Signup and view all the flashcards

Trigonometric Function Values

The outputs of trigonometric functions (sin, cos, tan, cot) for specific input values.

Signup and view all the flashcards

Monotonic Functions

Functions that always increase or always decrease over their entire domain.

Signup and view all the flashcards

Growth or Decay Processes

Situations where a quantity is increasing or decreasing over time.

Signup and view all the flashcards

Exponential and Logarithmic Functions for Data

These functions can be used to model real-world data that shows growth or decay.

Signup and view all the flashcards

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

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team

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.

More Like This

Properties of Functions Quiz
16 questions

Properties of Functions Quiz

CreativeAstrophysics4752 avatar
CreativeAstrophysics4752
أنواع العلاقات وخصائص الدوال
8 questions

أنواع العلاقات وخصائص الدوال

SpiritedMaroon9199 avatar
SpiritedMaroon9199
Function Types and Properties Quiz
9 questions

Function Types and Properties Quiz

StainlessJasper5137 avatar
StainlessJasper5137
Mathematics - Types of Relations and Functions
10 questions

Mathematics - Types of Relations and Functions

SuccessfulPopArt avatar
SuccessfulPopArt
Use Quizgecko on...
Browser
Open
Browser
Browser