"In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y."
A function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output.
Introduction to Functions: Understanding the concept of functions and how they are used to represent mathematical relationships between two or more variables.
Domain and Range: Defining the domain and range of a function and how to find them given a set of input and output values.
Function Notation: Understanding the notation used to represent a function and how to evaluate an expression given a particular input value.
Linear Functions: Learning about linear functions and how they can be represented using a variety of methods, including slope-intercept form, point-slope form, and standard form.
Quadratic Functions: Understanding the concept of quadratic functions and how to graph them using various methods such as completing the square, the quadratic formula, and factoring.
Polynomial Functions: Learning about polynomial functions of various degrees and their properties, including end behavior, roots, and extrema.
Rational Functions: Defining rational functions and studying their properties such as asymptotes, intercepts, and end behavior.
Exponential Functions: Learning about exponential functions, their properties, and how to graph them.
Logarithmic Functions: Understanding the concept of logarithmic functions, their properties, and how to graph them.
Trigonometric Functions: Learning about trigonometric functions such as sine, cosine, and tangent, and their properties, including amplitude, period, and phase shift.
Inverse Functions: Understanding the concept of inverse functions and how to find them given a function.
Composite Functions: Understanding the concept of composite functions and how to evaluate them.
Limits: Defining limits and understanding how they are used to study the behavior of functions as the input values approach a particular point.
Derivatives: Learning about derivatives, how they are defined, and how to find them using various methods such as the limit definition, power rule, and chain rule.
Applications of Derivatives: Understanding how derivatives can be used to study various properties of functions, including extrema, concavity, and optimization.
Integrals: Learning about integrals, how they are defined, and how to evaluate them using various methods such as integration by substitution, integration by parts, and partial fractions.
Applications of Integrals: Understanding how integrals can be used to find areas, volumes, and other quantities related to functions.
Sequences and Series: Learning about sequences and series, their properties, and how to find their sum, including infinite and geometric series.
Differential Equations: Understanding the concept of differential equations and how to solve them using various methods such as separation of variables, homogeneous equations, and first-order linear equations.
Multivariable Calculus: Learning about functions of several variables, partial derivatives, gradients, and how to find extrema of functions with more than one input variable.
Polynomial functions: A polynomial function is a function that is expressed as a sum of terms, where each term is a constant multiplied by one or more variables raised to a whole number exponent.
Rational functions: A rational function is a ratio of polynomials, where the denominator polynomial is not equal to zero.
Exponential functions: An exponential function is a function in which the independent variable appears as an exponent. The base of the exponent is a positive constant, usually denoted by the letter e.
Logarithmic functions: A logarithmic function is the inverse function of an exponential function. The logarithmic function has a base that is constant and different from 1.
Trigonometric functions: Trigonometric functions are functions that relate the angles of a right triangle to the lengths of its sides. The primary trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.
Hyperbolic functions: Hyperbolic functions are functions that are related to the hyperbola, a geometric shape similar to an ellipse. The primary hyperbolic functions are hyperbolic sine, hyperbolic cosine, and hyperbolic tangent.
Power functions: Power functions are functions that are expressed as a power of the independent variable. A power function can be represented in the form y = kx^n, where k and n are constants.
Piecewise functions: A piecewise function is a function that is defined on different parts of the domain using different expressions. A piecewise function can be discontinuous, meaning that there is a jump in the function at the point where the pieces meet.
Implicit functions: An implicit function is a function where the dependent variable is not explicitly defined in terms of the independent variable. Instead, the function is represented by an equation that includes both variables.
Circular functions: Circular functions are functions that can be defined in terms of circles. Examples of circular functions include the unit circle and trigonometric functions.
"The set X is called the domain of the function and the set Y is called the codomain of the function."
"Functions were originally the idealization of how a varying quantity depends on another quantity."
"For example, the position of a planet is a function of time."
"Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century."
"Until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity)."
"The concept of a function was formalized at the end of the 19th century in terms of set theory."
"A function is most often denoted by letters such as f, g and h."
"The value of a function f at an element x of its domain is denoted by f(x)."
"The numerical value resulting from the function evaluation at a particular input value is denoted by replacing x with this value."
"For example, the value of f at x = 4 is denoted by f(4)."
"When the function is not named and is represented by an expression E, the value of the function at, say, x = 4 may be denoted by E|x=4."
"For example, the value at 4 of the function that maps x to x^2 may be denoted by x^2|x=4."
"A function is uniquely represented by the set of all pairs (x, f(x)), called the graph of the function."
"The graph of the function [pairs (x, f(x))] is a popular means of illustrating the function."
"Functions are widely used in science, engineering, and in most fields of mathematics."
"It has been said that functions are 'the central objects of investigation' in most fields of mathematics."
"The concept of a function was formalized...greatly enlarging the domains of application of the concept."
"Functions are widely used in science, engineering, and in most fields of mathematics."
"Functions are 'the central objects of investigation' in most fields of mathematics."