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Functions

This course introduces functions which are the most common type of mathematical formula. It discusses the terms: function, domain, range, one-one function, inverse function, and composition of functions. It explains how to determine the domain and range of a function. It covers the use of the function notation. It also provides a detailed discussion about one-to-one and inverse functions. It also explains in words why a given function is a function or why it does not have an inverse. It also covers how to find the inverse of a one-one function and form composite functions, It explains how to use graphs to show the relationship between a function and its inverse.

6 topics

235 lectures

Educators

Course Curriculum

Introduction to Algebra
60 videos
Linear Functions
35 videos
Solve Linear Inequalities
35 videos
Functions
20 videos
Graph Linear Functions
40 videos
Write Linear Equations
45 videos

Functions Lectures

01:43
Functions

Function Notation - Example 1

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).
Julie Silva
03:18
Functions

Function Notation - Example 2

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).
Julie Silva
01:55
Functions

Function Notation - Example 3

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).
Julie Silva
01:50
Functions

Function Notation - Example 4

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).
Julie Silva
03:22
Functions

Function Notation - Overview

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).
Julie Silva
03:12
Functions

Graphing Functions by Making a Table - Example 1

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). The input and output of a function are often real numbers, but they can also be elements of any other set for which the relation makes sense. For instance, one can define a function from the set of integers to the set of even integers, or a function from the set of people to the set of people taller than six feet. Functions of various kinds appear in many areas of mathematics and science. There are infinitely many (algebraic) functions, such as the trigonometric functions sine and cosine, and exponential function. Functions also appear in the solutions of differential equations, in the study of differential geometry, and in many other areas of mathematics.
Julie Silva
02:27
Functions

Graphing Functions by Making a Table - Example 2

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). The input and output of a function are often real numbers, but they can also be elements of any other set for which the relation makes sense. For instance, one can define a function from the set of integers to the set of even integers, or a function from the set of people to the set of people taller than six feet. Functions of various kinds appear in many areas of mathematics and science. There are infinitely many (algebraic) functions, such as the trigonometric functions sine and cosine, and exponential function. Functions also appear in the solutions of differential equations, in the study of differential geometry, and in many other areas of mathematics.
Julie Silva
02:33
Functions

Graphing Functions by Making a Table - Example 3

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). The input and output of a function are often real numbers, but they can also be elements of any other set for which the relation makes sense. For instance, one can define a function from the set of integers to the set of even integers, or a function from the set of people to the set of people taller than six feet. Functions of various kinds appear in many areas of mathematics and science. There are infinitely many (algebraic) functions, such as the trigonometric functions sine and cosine, and exponential function. Functions also appear in the solutions of differential equations, in the study of differential geometry, and in many other areas of mathematics.
Julie Silva
03:18
Functions

Graphing Functions by Making a Table - Example 4

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). The input and output of a function are often real numbers, but they can also be elements of any other set for which the relation makes sense. For instance, one can define a function from the set of integers to the set of even integers, or a function from the set of people to the set of people taller than six feet. Functions of various kinds appear in many areas of mathematics and science. There are infinitely many (algebraic) functions, such as the trigonometric functions sine and cosine, and exponential function. Functions also appear in the solutions of differential equations, in the study of differential geometry, and in many other areas of mathematics.
Julie Silva
03:36
Functions

Graphing Functions by Making a Table - Overview

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). The input and output of a function are often real numbers, but they can also be elements of any other set for which the relation makes sense. For instance, one can define a function from the set of integers to the set of even integers, or a function from the set of people to the set of people taller than six feet. Functions of various kinds appear in many areas of mathematics and science. There are infinitely many (algebraic) functions, such as the trigonometric functions sine and cosine, and exponential function. Functions also appear in the solutions of differential equations, in the study of differential geometry, and in many other areas of mathematics.
Julie Silva
02:52
Functions

Relations and Functions - Example 1

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. In many other cases, a function is given by a picture, called the graph of the function. Other functions may be defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.
Julie Silva
01:43
Functions

Relations and Functions - Example 2

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. In many other cases, a function is given by a picture, called the graph of the function. Other functions may be defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.
Julie Silva
02:45
Functions

Relations and Functions - Example 3

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. In many other cases, a function is given by a picture, called the graph of the function. Other functions may be defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.
Julie Silva
02:40
Functions

Relations and Functions - Example 4

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. In many other cases, a function is given by a picture, called the graph of the function. Other functions may be defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.
Julie Silva
12:24
Functions

Relations and Functions - Overview

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x). Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. In many other cases, a function is given by a picture, called the graph of the function. Other functions may be defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.
Julie Silva
02:36
Functions

The Coordinate Plane - Example 1

In mathematics, the Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The origin (0,0) is usually located at the center of the grid formed by the points, but in some contexts (such as graphing) it is useful to define the coordinates so that the origin is "outside" the grid, allowing the coordinates to be interpreted as relative distances from the origin. The two perpendicular axes are called the "axes" of the Cartesian coordinate system, and the points where they meet are its "intercepts". The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
Julie Silva
02:51
Functions

The Coordinate Plane - Example 2

In mathematics, the Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The origin (0,0) is usually located at the center of the grid formed by the points, but in some contexts (such as graphing) it is useful to define the coordinates so that the origin is "outside" the grid, allowing the coordinates to be interpreted as relative distances from the origin. The two perpendicular axes are called the "axes" of the Cartesian coordinate system, and the points where they meet are its "intercepts". The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
Julie Silva
02:50
Functions

The Coordinate Plane - Example 3

In mathematics, the Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The origin (0,0) is usually located at the center of the grid formed by the points, but in some contexts (such as graphing) it is useful to define the coordinates so that the origin is "outside" the grid, allowing the coordinates to be interpreted as relative distances from the origin. The two perpendicular axes are called the "axes" of the Cartesian coordinate system, and the points where they meet are its "intercepts". The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
Julie Silva
05:07
Functions

The Coordinate Plane - Example 4

In mathematics, the Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The origin (0,0) is usually located at the center of the grid formed by the points, but in some contexts (such as graphing) it is useful to define the coordinates so that the origin is "outside" the grid, allowing the coordinates to be interpreted as relative distances from the origin. The two perpendicular axes are called the "axes" of the Cartesian coordinate system, and the points where they meet are its "intercepts". The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
Julie Silva
05:43
Functions

The Coordinate Plane - Overview

In mathematics, the Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The origin (0,0) is usually located at the center of the grid formed by the points, but in some contexts (such as graphing) it is useful to define the coordinates so that the origin is "outside" the grid, allowing the coordinates to be interpreted as relative distances from the origin. The two perpendicular axes are called the "axes" of the Cartesian coordinate system, and the points where they meet are its "intercepts". The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
Julie Silva

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