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The document discusses functions and function notation. It defines a function as a special type of relation where each element of the independent variable is paired with exactly one element of the dependent variable. It provides examples of different ways to represent functions, including coordinate graphs, symbolic rules, and ordered pairs. The document also discusses evaluating functions, the domain and range of functions, and identifies whether certain relations are functions or not.
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Functional dependency defines a relationship between attributes in a table where a set of attributes determine another attribute. There are different types of functional dependencies including trivial, non-trivial, multivalued, and transitive. An example given is a student table with attributes Stu_Id, Stu_Name, Stu_Age which has the functional dependency of Stu_Id->Stu_Name since the student ID uniquely identifies the student name.
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This document lists and defines vocabulary words from math lessons on various topics:
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- Rational functions are functions whose rules can be written as a ratio of two polynomials. These may be continuous or discontinuous.
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▹HISTORY
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▹LIST OF COMMON INDETERMINATE FORMS
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combination of functions in an independent variable may often be evaluated by
replacing these functions by their limits.
If the expression obtained after this substitution does not give enough information to
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In DBMS (DataBase Management System), the relation algebra is important term to further understand the queries in SQL (Structured Query Language) database system. In it just give up the overview of operators in DBMS two of one method relational algebra used and another name is relational calculus.
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CONTENTS:-
▹HISTORY
▹DEFINITION
▹TYPES OF INDETERMINATE FORMS
▹L-HOSPITAL’S RULE
▹LIST OF COMMON INDETERMINATE FORMS
▹
▹
-HISTORYThe
term was originally introduced by Cauchy's student Moigno (also known as
François-Napoléon-Marie Moigno) in the middle of the 19th century.
▹
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What are indeterminate forms?
In calculus and other branches of mathematical analysis, limits involving an algebraic
combination of functions in an independent variable may often be evaluated by
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This document is a student assignment on joins and their types in database management systems. It defines joins as combining related tuples from two relations based on matching conditions. The main types of joins discussed are inner joins (theta, equi, natural), and outer joins (left, right, full). Inner joins return only tuples that satisfy the condition, while outer joins return all tuples from one or both relations whether or not they match. Examples are provided to illustrate each join type.
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An Open Innovation Workshop as part of our FP7 SME-AERO-POWER Project. Facilitated by Ron Donaldson of Argenta Europ Ltd and Hosted by Óscar Sacristán de Frutos of Cidault. We took a group of mostly SMEs through a series of workshop methods aimed at amplifying networking, collaboration and the production of opportunities for innovation to their particular innovation ecosystem.
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Similar to Lesson 2 1
Lesson 1
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The domain is the set of all possible x-values, and the range is the resulting y-values. The rate of change of a function can be found using the difference quotient or slope formula. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point. Domain restrictions limit the possible input values of a function. Horizontal and vertical asymptotes provide information about the behavior of a rational function as x approaches positive or negative infinity.
Lesson 1
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The domain is the set of all possible x-values, and the range is the resulting y-values. To find the rate of change, use the difference quotient which is similar to the slope formula. The vertical line test determines if a graph represents a function - if any vertical line intersects the graph more than once, it is not a function. Domain restrictions specify allowed input values.
Edsc 304 lesson 1
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The rate of change of a function is found by taking the difference quotient, which is similar to the slope formula. The domain of a function is the set of all possible x-values, while the range is the resulting y-values. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point.
function
1) Functions relate inputs to outputs through ordered pairs where each input maps to exactly one output. The domain is the set of inputs and the range is the set of outputs.
2) There are different types of functions including linear, quadratic, and composition functions. A linear function's graph is a straight line while a quadratic function's graph is a parabola.
3) Composition functions combine other functions where the output of one becomes the input of another. Together functions provide a powerful modeling tool used across many fields including medicine.
2.3 Functions
This document defines and discusses functions. It begins by defining a relation and function, noting that a function is a special type of relation where each input is mapped to exactly one output. It introduces function notation and discusses the domain, codomain, and range of a function. Examples are provided to illustrate determining if a relation defines a function. The document also covers identifying functions from equations or graphs, and the vertical line test. It concludes with a discussion of function notation and classwork assignments.
Lecture co3 math21-1
This document discusses concepts related to calculus including limits, continuity, and derivatives of functions. Specifically, it covers:
- Definitions and theorems related to limits, continuity, and derivatives of algebraic functions.
- Evaluating limits, determining continuity of functions, and taking derivatives of algebraic functions using basic theorems of differentiation.
- The objective is for students to be able to evaluate limits, determine continuity, and find derivatives of continuous algebraic functions in explicit or implicit form after discussing these calculus concepts.
4.6 Relations And Functions
The document defines relations and functions. A relation is a set of ordered pairs where the domain is the set of x-values and the range is the set of y-values. A function is a relation where each domain value is paired with exactly one range value. The document provides examples of determining if a relation represents a function using the vertical line test and evaluating functions using function notation.
Relations and Functions
The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
Special Topics on Functions, Sequences, and Series MAT117 .docx
Special Topics on Functions, Sequences, and Series
MAT/117 Version 9
1
Copyright © 2013 by University of Phoenix. All rights reserved.
University of Phoenix Material
Special Topics on Functions, Sequences, and Series
The goal of this week is to introduce more concepts and skills about functions that you started learning in
the previous week. In addition, you will learn to identify when a list of numbers have an arithmetic or
geometric pattern.
Composition of Functions
The composition of functions is denoted with the notation .
This operation consists on the idea that the input of one function is another function, or algebraically, this
will be written as . This means “The input of is .”
In Cognitive Tutor, you will learn composition of linear functions. However, it is possible to do composition
of functions with any type of functions. Here is an example between linear and quadratic functions:
Example
Let and – .
So,
( )
Careful: is not the same as . Order matters! See the example below using the same
functions above.
( )
One-to-One Functions
Previously, you learned that the basic idea of a function is that “every input has exactly one output,” or
“every x value has exactly one y value.”
For example, in the graph of the function below, you can see every x value corresponds with (matches)
exactly one y value (look closely how each vertical gridline intersects the graph only once; this is called
the Vertical Line Test).
Special Topics on Functions, Sequences, and Series
MAT/117 Version 9
2
Copyright © 2013 by University of Phoenix. All rights reserved.
If you look closely the graph above, there are different inputs that have the same output (or there are
different x values that correspond to the same y value).
Now, when each input of a function has a different output (or every x value has a different y value), then
we say the function has a one-to-one correspondence. Here is an example of a one-to-one function:
In the graph above, every x value has a different y value (look closely how each horizontal gridline
intersects the graph only once; this is called the Horizontal Line Test).
Understanding one-to-one functions is essential to understand inverse functions, which you will learn in
Cognitive Tutor. A real-life example of one-to-one functions is the following: Every person has a unique
social security number and each social security number corresponds to a person.
Exponential Functions
When a constant (a fixed number) is raised to a variable (say, x), we are working with exponential
functions. Algebraically, exponential functions have the form , where a > 0 but cannot be 1.
For example, . When the value of is greater than 1, the graph of these functions have this
form:
From the graph above, as ...
relationsandfunctionsupdated-140102120840-phpapp01.ppt
This document discusses relations and functions. It provides examples of how to classify relations as functions or not based on whether they satisfy the vertical line test and one-to-one correspondence. Students are asked to complete activities identifying domains and ranges of relations, graphing relations to apply the vertical line test, and determining if equations represent functions. The assignment is to copy equations on a sheet of paper and identify if they represent functions.
math 8 kumustahan.pptx
This document defines and provides examples of relations and functions. It explains that a relation is a set of ordered pairs where the first elements are the domain and second elements are the range. Relations can be represented as tables, mapping diagrams, graphs or rules. A function is a special type of relation where each domain element is mapped to only one range element. The document provides examples of determining if a set of ordered pairs or graph represent a function. It also defines independent and dependent variables, and explains that the domain is the set of independent inputs and the range is the set of dependent outputs.
2.1 Basics of Functions and Their Graphs
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
Algebra 2 Section 2-1
This document provides an overview of functions and continuity. It begins with essential questions about determining if functions are one-to-one and/or onto, and determining if functions are discrete or continuous. The document then defines key vocabulary terms related to functions, including one-to-one functions, onto functions, discrete relations, continuous relations, and more. It provides examples to demonstrate these concepts, such as evaluating functions, graphing equations, and determining if a relation represents a function.
Chapter on Functions and Graphs.ppt
A relation is a set of ordered pairs. A function is a relation where each domain value has only one range value. To determine if a relation is a function, use the vertical line test or check if each x-value only has one y-value. An equation defines a function if each x only corresponds to one y when solving the equation for y. Piecewise functions are defined by two or more equations over different parts of the domain. The slope of a line is rise over run and can be found by calculating change in y over change in x between any two points. You can write the equation of a line from its point-slope form or by finding the slope between two points and plugging into point-slope form
LANGUAGE-OF-RELATIONS-AND-FUNCTIONS.pptx
This document discusses relations and functions in mathematics. It defines relations as sets of ordered pairs that show relationships between inputs and outputs, while functions are special relations where each input is mapped to only one output. The document provides examples of relations and functions, and examples of domain, range, and how to determine if a relation is a function. It concludes with a practice problem asking the reader to identify the domain, range, and whether relations are functions.
Chapter05
This document discusses summarizing bivariate data using scatterplots and correlation. It provides an example of fare data from a bus company that is modeled using linear and nonlinear regression. Linear regression finds a strong positive correlation between distance and fare, but the relationship is better modeled nonlinearly using the logarithm of distance. The nonlinear model accounts for 96.9% of variation in fares compared to 84.9% for the linear model.
3.1 Functions and Function Notation
* Determine whether a relation represents a function.
* Find the value of a function.
* Determine whether a function is one-to-one.
* Use the vertical line test to identify functions.
* Graph the functions listed in the library of functions.
Relations and functions
The document provides information about relations and functions:
- It defines key terms like relation, domain, range, ordered pair, and function.
- It explains how to represent relations using ordered pairs, tables, graphs, and mappings.
- It discusses how to determine if a relation is a function using the vertical line test or by checking if each domain value is mapped to only one range value.
- It distinguishes between independent and dependent variables in functional notation.
Variables and patterns vocabulary
The document contains definitions and examples for vocabulary terms related to variables and patterns. There are definitions for variable, independent variable, dependent variable, coordinate pair, x-axis, y-axis, scale, relationship, equation, pattern, rule, table, coordinate, change, income, cost, and profit. Examples are provided to illustrate each term. The document also outlines the four steps for making coordinate graphs and indicates that vocabulary notes are ending and an investigation will begin.
Normalization
The document discusses database normalization and functional dependencies. It defines normalization as imposing rules on database tables to eliminate anomalies during data manipulation. Functional dependency is defined as a relationship where a set of attributes determines another. The properties of functional dependencies - reflexive, augmentation, transitive, union, and decomposition - are explained and examples are given. Normalization and understanding functional dependencies helps design high quality databases without redundancies or anomalies.
Similar to Lesson 2 1 (20)
Special Topics on Functions, Sequences, and Series MAT117 .docx
Special Topics on Functions, Sequences, and Series MAT117 .docx
relationsandfunctionsupdated-140102120840-phpapp01.ppt
relationsandfunctionsupdated-140102120840-phpapp01.ppt
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Lesson 2-5
The document compares exponential and linear models. Exponential models exhibit constant percentage change determined by the growth factor b, while linear models show constant increase/decrease set by the slope m. Exponential equations can be determined from a single point, unlike linear equations which require two points to find the parameters m and b. The document demonstrates finding the exponential equation for two given points (2,6) and (4,10) by setting up equations and dividing them to isolate the growth factor b.
Lesson 2 2
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Section 2 3 (1)
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Section 2.5 (2)
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Section 2.4
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Lesson 1 7 (3)
Variance and standard deviation are measures of spread used to describe the shape of distributions associated with the mean. While two distributions may have the same mean, their variance and standard deviation can show that they have very different shapes. To calculate variance and standard deviation, you first find the deviation of each value from the mean, square the deviations, and sum them. You then divide the sum by n-1 to get the variance, and take the square root of the variance to find the standard deviation.
Section 1-5
The document discusses the concept of limits in calculus, including the formal definition of a limit as the value a function approaches as the input gets arbitrarily close to a given number. It provides examples of evaluating limits graphically and algebraically, such as determining the limit of a function as x approaches 3 is 6, even though the function is undefined at x=3 due to a removable discontinuity. The document also discusses using graphs to identify different types of discontinuities and limits.
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The document discusses using the trapezoidal rule to evaluate definite integrals by approximating the area under a curve between bounds with trapezoids. It explains that the trapezoidal rule is a more exact method than counting squares. To use the rule, the integral bounds are divided into trapezoids, the function is evaluated at the endpoints, and the areas are calculated by multiplying the average base by the height and summing the results, with endpoints weighted by one half.
Lesson 1-4 -- Five Number Summary
The document discusses the five number summary, which includes the minimum, first quartile, median, third quartile, and maximum values. It provides an example of the heights of a basketball team, calculates their five number summary and interquartile range, and discusses how outliers can be identified using 1.5 times the interquartile range.
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The document discusses using integrals to calculate distance traveled based on a velocity-time graph. It explains that the area under the graph between two times represents the distance traveled during that interval. Specifically, it asks the reader to:
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2) Count the squares and find the distance traveled from 0-20 seconds by finding the area under the curved graph.
3) Estimate the definite integral of a function representing a football's cross-sectional area to find its total volume.
Lesson 1-3
The document discusses different measures of central tendency, including the mean and median. It introduces the concept of using sigma notation to simplify writing out long sums, such as when calculating the mean of a data set. It provides an example of entree prices from a restaurant and encourages writing out the formula to calculate the mean using sigma notation.
Section 1-2
This document discusses functions, their derivatives, and rates of change. It introduces the concept of the derivative as the instantaneous rate of change of a function. It then asks the reader to sketch and describe different types of functions without a calculator. It provides a sample graph showing sections that are increasing, decreasing, or neither, and asks the reader to identify these sections and determine where the rate of change is more or less quick.
Lesson 1 2
A stemplot is a method to display data by splitting each number into a stem and leaf, with the ones digit as the leaf and other digits as the stem. It allows comparison of two data sets side by side. A dotplot displays the frequency of data values and is useful when the data is categorical. It may not be suitable to display continuous numeric data like golf scores.
5.12.08 Advanced Factoring1
Factoring by grouping allows factoring out the greatest common factor from some but not all terms of a polynomial, which can make the entire factorization easier. This technique can be used to factor difficult quadratic trinomials. Examples are provided to illustrate factoring by grouping.
5.6.08 Fundamental Theorem Of Algebra1
The fundamental theorem of algebra states that any polynomial of degree greater than 1 with complex coefficients has at least one complex zero. A related theorem is that a polynomial of degree n has at most n zeros. Another theorem is that a polynomial of degree greater than 1 with complex coefficients has exactly n complex zeros counting multiplicities.
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4.9.08 Polynomial Models3
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4.10.08 Graphs Of Polynomials
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Pascal's Triangle is an array of numbers arranged in triangular form where each number is derived from adding the two numbers directly above it. To generate the triangle, a chart is made where the terms are found by taking the binomial coefficients of n over r, and properties include that each row has one more term than the previous row and the first and last terms of each row are always 1.
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Lesson 2 1
- 1. Functions and Function Notation Any set of ordered pairs is known as a relation . A relation consists of an independent variable and a dependent variable . Usually, we think of x as the independent variable, and y as the dependent variable. The relation to the left consists of 6 ordered pairs. A function is a special kind of relations, in which each member of the independent variable is paired with one and only one value of the dependent variable. Is the relation to the left a function? Another way to state the definition of a function is that each member of the domain corresponds with exactly one member of the range . What is meant by domain and range?
- 2. Functions and Function Notation There are many ways to represent a function. Here are examples of three of the most common: A coordinate graph: A rule expressed symbolically or in words: “ The sale price is 20% off of the sticker price.” A list of ordered pairs: (3,5), (-1,8), (9,9), (-12, -1) -2 5 -5 0 1 1 6 4 -2 3 y x
- 3. Functions and Function Notation You have all seen function notation at some point in the past: If f ( x ) is defined as above, how would you evaluate f (-3)? How would you evaluate f (a+b)? f (watermelon)?
- 4. Are all relations functions? Consider the set of all points that are 3 units away from the point (2,2). What does a coordinate graph representing those points look like? What are the domain and range of this relation? This relation is not a function because for some members of the domain ( x values), there is more than one corresponding value in the range ( y values). You may recall the vertical line test .