Functions ordered pairs
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This document defines and provides examples of functions. It explains that a function is a relation where each element in the domain is paired with exactly one element in the range. It also notes that visually, a relation is a function if it passes the vertical line test, meaning a vertical line can only intersect the graph once. Examples of ordered pairs and relations that are and are not functions are given to illustrate these concepts.
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Math functions, relations, domain & range
This document discusses math functions, relations, and their domains and ranges. It provides examples of relations and explains that the domain is the set of first numbers in ordered pairs, while the range is the set of second numbers. A function is defined as a relation where each x-value has only one corresponding y-value. It compares examples of relations that are and are not functions and describes how to evaluate functions by inserting values. Tests for identifying functions like the vertical line test are also outlined.
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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.
Math presentation on domain and range
This document defines key concepts related to functions including domain, co-domain, and range. It provides examples of determining the domain and range of various functions. The domain of a function is the set of inputs, the co-domain is the set of all possible outputs, and the range is the set of actual outputs. Examples show how to use inequalities to find the domain by determining when a function gives real values and how to manipulate equations to find the range.
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The document discusses various factoring techniques, including:
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- Factoring trinomials of the form ax^2 + bx + c by finding two numbers that multiply to c and add to b
- Factoring the difference of two squares and perfect square trinomials
- Using the "box method" to factor trinomials by placing factors of the first and last terms in the boxes
- Factoring by grouping polynomials with four terms into two groups of two terms each.
Joint variation final
Here are the answers to the activity questions on page 216 of the study guide in one whole sheet of paper:
5. Kinetic energy (KE) varies jointly as mass (m) and the square of velocity (v2).
KE = kmv2
Given:
Mass (m) = 8 g
Velocity (v) = 5 cm/s
KE = 100 ergs
Find k:
100 ergs = k(8 g)(5 cm/s)2
100 ergs = k(8)(25)
100 ergs = 200k
k = 100/200 = 1/2
KE = kmv2
Given:
Mass (m) = 6 g
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This document discusses direct and inverse proportion. It defines direct proportion as two quantities increasing or decreasing together such that their ratio remains constant. An example shows how to calculate the amount of sugar needed given the amount of water. Inverse proportion means one quantity increases as the other decreases, keeping their product constant. An example calculates how long food will last for additional hens. Direct proportion uses a formula of x/y=k while inverse proportion uses xy=k. Examples are provided to demonstrate calculating unknown values using these formulas.
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Math functions, relations, domain & range
This document discusses math functions, relations, and their domains and ranges. It provides examples of relations and explains that the domain is the set of first numbers in ordered pairs, while the range is the set of second numbers. A function is defined as a relation where each x-value has only one corresponding y-value. It compares examples of relations that are and are not functions and describes how to evaluate functions by inserting values. Tests for identifying functions like the vertical line test are also outlined.
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.
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This document defines key concepts related to functions including domain, co-domain, and range. It provides examples of determining the domain and range of various functions. The domain of a function is the set of inputs, the co-domain is the set of all possible outputs, and the range is the set of actual outputs. Examples show how to use inequalities to find the domain by determining when a function gives real values and how to manipulate equations to find the range.
Factoring
The document discusses various factoring techniques, including:
- Factoring by greatest common factor (GCF)
- Factoring trinomials of the form ax^2 + bx + c by finding two numbers that multiply to c and add to b
- Factoring the difference of two squares and perfect square trinomials
- Using the "box method" to factor trinomials by placing factors of the first and last terms in the boxes
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Joint variation final
Here are the answers to the activity questions on page 216 of the study guide in one whole sheet of paper:
5. Kinetic energy (KE) varies jointly as mass (m) and the square of velocity (v2).
KE = kmv2
Given:
Mass (m) = 8 g
Velocity (v) = 5 cm/s
KE = 100 ergs
Find k:
100 ergs = k(8 g)(5 cm/s)2
100 ergs = k(8)(25)
100 ergs = 200k
k = 100/200 = 1/2
KE = kmv2
Given:
Mass (m) = 6 g
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This slide introduce the concept of mathematical function as well as the concepts of domain and range.
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The document discusses the relationship between the remainder when dividing a function f(x) by (x-c) and the value of f(c). Specifically, it states that the remainder is equal to the value of f(c), and if the remainder is 0, then (x-c) is a factor of the function. This relationship means that using synthetic division to find the remainder can simplify calculations, as the remainder directly provides the value of the function at that point. The document also includes a lame joke about French fries.
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Here are the steps to solve each inequality and graph the solution:
1) m + 14 < 4
-14 -14
m < -10
Graph: m < -10
2) -7 > y-1
+1 +1
-6 > y
y < -6
Graph: y < -6
3) (-3)k < 10(-3)
-3
k > -30
Graph: k > -30
4) 2x + 5 ≤ x + 1
-x -x
x + 5 ≤ 1
-5 -5
x ≤ -4
Graph: x ≤ -4
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3) Solve the remaining polynomial equation
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The document provides information about functions and relations. It defines a function as a relation where each x-value is paired with exactly one y-value. To determine if a relation is a function, it describes using the vertical line test, where a relation is a function if a vertical line can only intersect the graph at one point. It gives examples of applying the vertical line test to graphs and determining the domain and range of relations.
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This document discusses functions and relations. It defines a relation as a set of ordered pairs and provides examples. It then defines a function as a special type of relation where each element of the domain corresponds to exactly one element of the range, meaning no two ordered pairs can have the same first element. The document discusses different types of relations including one-to-one, one-to-many, many-to-one, and many-to-many. It also discusses how functions can be presented using arrow diagrams, tables, graphs, and ordered pairs. Finally, it discusses function notation and evaluating functions by substituting values into the function.
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The document discusses finding the domain and range of functions. It defines domain as the set of possible input values and range as the set of possible output values. It provides examples of determining the domain from graphs, equations, and piecewise functions by identifying restrictions on inputs. Interval and set-builder notation are introduced for writing domains and ranges. Determining whether values are included or excluded based on open or closed intervals is also covered.
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Inverse Functions
1) The document discusses finding the inverse of functions by interchanging the x and y variables and solving for y. It provides examples of finding the inverses of f(x)=3x-7 and g(x)=2x^3+1.
2) It also discusses verifying inverses by checking if the composition of a function and its inverse equals x. And finding inverses of functions with restricted domains, including an example of f(x)=sqrt(x+4).
3) Finally, it discusses the relationship between a function being one-to-one and having an inverse function, both algebraically and graphically.
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1. A function is a relation where each input is paired with exactly one output.
2. To determine if a relation is a function, use the vertical line test - if any vertical line intersects more than one point, it is not a function.
3. To find the value of a function, substitute the given value for x into the function equation and simplify.
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* IGCSE Practice Revision Exercise which covers all the related concepts required for students to unravel any IGCSE Exam Style Transformation Questions
* Learner will be able to say authoritatively that:
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Logarithms were invented to solve exponential equations like 2x = 6, where x is between 2 and 3. A logarithm with base b is defined as logby = x if and only if bx = y. Common properties of logarithms include logb1 = 0 and logbb = 1. Logarithmic functions have inverse relationships with exponential functions and can be graphed by shifting and reflecting the standard logarithmic curve.
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The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant other than 1. It discusses how the graph of an exponential function depends on whether b is greater than or less than 1. Specifically, if b > 1 the graph increases to the right, and if 0 < b < 1 the graph decreases to the right. The document also covers transformations of exponential functions, including vertical and horizontal shifting, reflecting, and stretching/shrinking. It introduces the special number e, defines it as the limit of (1 + 1/n)n as n approaches infinity, and discusses its role in compound interest formulas.
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The document discusses index notation and how it is used to represent repeated multiplication. It covers the basic rules for multiplying and dividing terms with the same base, including adding/subtracting the indices. It also discusses zero and negative indices, and how numbers raised to the power of 0 or negative powers can be evaluated. Key rules covered are a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), a^0 = 1, and a^-n = 1/a^n.
Inequalities ppt revised
Here are the steps to solve each inequality and graph the solution:
1) m + 14 < 4
-14 -14
m < -10
Graph: m < -10
2) -7 > y-1
+1 +1
-6 > y
y < -6
Graph: y < -6
3) (-3)k < 10(-3)
-3
k > -30
Graph: k > -30
4) 2x + 5 ≤ x + 1
-x -x
x + 5 ≤ 1
-5 -5
x ≤ -4
Graph: x ≤ -4
Piecewise functions
The document discusses piecewise functions, which are functions defined by multiple rules over different parts of the domain. It provides examples of piecewise functions, including how to graph them by applying each rule over the appropriate portion of the domain. It also discusses evaluating piecewise functions at given values, writing piecewise functions based on graphs, and analyzing properties like extrema. Piecewise functions allow modeling of real-world situations using multiple equations each corresponding to a part of the input range.
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Exponential functions
The document discusses exponential functions and exponential equations. Exponential functions have the form f(x) = bx, where b is the base and x is the exponent. These functions are important in modeling real-world phenomena like population growth. Exponential equations set the exponents of the same base equal to solve for the variable. They can be solved by rewriting all terms to have the same base, setting the exponents equal, and solving the resulting equation.
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Rational equations are equations where a variable occurs in the denominator of a rational expression. To solve rational equations:
1) Identify values that result in no solution by finding values where the denominator equals 0.
2) Find the lowest common denominator (LCD) and multiply both sides of the equation by the LCD.
3) Solve the numerators and reject any solutions that result in the denominator equaling 0.
Inverse functions
The document discusses inverse functions and how an inverse function undoes the operations of the original function. It provides examples of finding the inverse of functions by switching the x and y values and solving for y. The inverse of a function will be a function itself only if the original function passes the horizontal line test.
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This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
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Lesson 13: Exponential and Logarithmic Functions (slides)
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The document provides information about functions and relations. It defines a function as a relation where each x-value is paired with exactly one y-value. To determine if a relation is a function, it describes using the vertical line test, where a relation is a function if a vertical line can only intersect the graph at one point. It gives examples of applying the vertical line test to graphs and determining the domain and range of relations.
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This document discusses functions and relations. It defines a relation as a set of ordered pairs and provides examples. It then defines a function as a special type of relation where each element of the domain corresponds to exactly one element of the range, meaning no two ordered pairs can have the same first element. The document discusses different types of relations including one-to-one, one-to-many, many-to-one, and many-to-many. It also discusses how functions can be presented using arrow diagrams, tables, graphs, and ordered pairs. Finally, it discusses function notation and evaluating functions by substituting values into the function.
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Objective 1 - Identifying Functions
Determine if a mathematical relation represented by a schematic map, a table of values, or a set of ordered pairs is or is not a function.
Objective 1 - Identifying Functions
Determine if a mathematical relation, represented by a schematic map, a table of values, or a set of ordered pairs, is or is not a function
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The document defines relations and functions. A relation is a set of ordered pairs where the domain is the set of first coordinates and the range is the set of second coordinates. A function is a special type of relation where each element in the domain corresponds to exactly one element in the range. Functions can be represented as tables of values, ordered pairs, mappings, graphs or equations. The vertical line test can be used to determine if a graph represents a function - if no vertical line intersects the graph at more than one point, it is a function. Examples are provided to illustrate these concepts.
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This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
Lesson 2 - Functions and their Graphs - NOTES.ppt
The document provides lesson material on functions and their graphs. It includes sections on defining functions, determining if a relation is a function, functional notation, domain and range, graphing functions, and identifying intervals of increase/decrease. Additional topics covered are relative min/max values, step functions, even and odd functions, and piecewise-defined functions. Examples and exercises are provided to illustrate key concepts.
Functions
A function is a relation where each element of the domain is paired with exactly one element of the range. The key characteristics are that there is one and only one output for each input. Some examples are provided to demonstrate determining if a relation represents a function based on this definition. The vertical line test is also introduced as a visual way to determine if a graph depicts a function. Finally, examples are given of evaluating functions for given inputs using function notation.
Functions and Relations
The document discusses functions and operations with functions. It defines a function as a relation where each input has a unique output. It introduces function notation using f(x) to represent the output of a function f given an input x. It then discusses how to evaluate functions using this notation. Finally, it discusses how functions can be combined using the basic operations of addition, subtraction, multiplication and division to create new functions.
Module 4 topic 1 part 3 notes
This document defines key terms related to relations and functions, including:
- A relation is a set of ordered pairs. The domain is the set of x-values and the range is the set of y-values.
- A relation is a function if each x-value is mapped to only one y-value. This can be determined using the vertical line test or by checking if x-values are repeated.
- Graphs where a vertical line crosses the relation more than once are not functions, while graphs where it only crosses once are functions.
A1 11 functions
The document discusses functions, their domains and ranges, function notation, evaluating functions, and sequences. It provides examples of arithmetic and geometric sequences and how they can be defined using equations. It also discusses determining whether numbers are in the domain of a function and finding the domain and range of functions given their equations or graphs.
Relations and Functions
The document defines relations and functions. A relation is a set of ordered pairs, while a function is a special type of relation where each x-value is mapped to only one y-value. The domain is the set of x-values and the range is the set of y-values. Functions can be identified using the vertical line test or by mapping the relation to check if any x-values are mapped to multiple y-values. Evaluating functions involves substituting domain values into the function rule to find the corresponding range values.
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Functions ordered pairs
- 1. Aim: What is a function? Do Now: Place your contracts in the basket. Take out a pen or pencil and a sheet of paper. You have 50 minutes to complete the diagnostic. This counts for Quiz 1. You will not be graded on correctness. Your grade will depend on how many questions you answer and how much work you show.
- 2. Functions as a set of ordered pairs. Domain: The set of all inputs in a function, or the set of x values. Range: The set of all outputs of a function, or the set of y values. DEFINITIONS:
- 3. A function is a relation in which each element of the domain is paired with exactly one element in the range. Example: {(1, 1), (2, 4), (3, 9), (4, 16)} Domain Range Mapping Diagram
- 4. Ordered pairs on the grid . . . . Write the ordered pairs for the relation shown in the graph. What is the Domain? (x values) What is the Range? (y values) Is this relation a function? Domain Range
- 5. Is the relation {(1, 3), (4, 3), (-2, -4),(0, -5)} a function or not? Why or why not? Domain Range . . . .
- 6. Function or not a function? {(0, 1), (2, 2), (0, 3), (4, 5)} Domain Range Not a function because the same value in the domain is paired with two different values in the range.
- 7. Visually, a relation is a function if it passes the vertical line test. When a vertical line is drawn if it only hits the graph of a relation once, then the relation is a function. ✔ ✖
- 8. Are these graphs of functions?
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