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function and relation worksheet with answer key

There is exactly one output for every input. We can visually identify functions by their graphs using the vertical line test15. 4. endobj P 0000000943 00000 n The relation depicts the connection between input and output. Given \(g ( x ) = x ^ { 2 }\), find \(g (2), g ( \frac{1}{2} )\), and \(g (x + h)\). .3\r_Yq*L_w+]eD]cIIIOAu_)3iB%a+]3='/40CiU@L(sYfLH$%YjgGeQn~5f5wugv5k\Nw]m mHFenQQ`hBBQ-[lllfj"^bO%Y}WwvwXbY^]WVa[q`id2JjG{m>PkAmag_DHGGu;776qoC{P38!9-?|gK9w~B:Wt>^rUg9];}}_~imp}]/}.{^=}^?z8hc' {(3,2), (1, -1), (2, -4), (3, -9), (4, -16)} 4. . )feOJeB_~n We can see that \(g(x)=2\) where \(x=5\); in other words, \(g(5)=2\). Here we separate the domain (x-values), and the range (y-values), and depict the correspondence between the values with arrows. Find \(x\) where \(g (x) = 5, g (x) = 0\), and \(g (x) = 10\). Determine the domain and range of the following relation and state whether it is a function or not: \(\{(1, 4), (0, 7), (2, 3), (3, 3), (4, 2)\}\). These cards are most appropriate for Math 8-Algebra 1.Task cards are very versatile, and can, This r squared creation is the complementary worksheet for our 4.2 PowerPoint, which students will complete as homework. << 8 0 obj Often, we can determine the domain and range of a relation if we are given its graph. /Creator () A function, on the other hand, is a relation that produces one output for each given input. This is a relations and functions worksheet. stream The relation is a function. 4. Free worksheet(pdf) and answer key on distinguishing functions from relations, stating domain and range and more. Research and discuss the life and contributions of Ren Descartes. /MediaBox [0 0 612.000000 792.000000] In addition, since we can find a vertical line that intersects the graph more than once, we conclude that the graph is not a function. Some filters moved to Formats filters, which is at the top of the page. What was the value of the car when it was new in \(1970\)? 0000000750 00000 n In the following exercises, evaluate the function: a. g(h2) b. g(x + 2) c. g(x) + g(2). This has been SO helpful when im stuck on problems that i just can't understand. stream Worksheet 4.1 relations and functions answer key. /Producer ( Q t 5 . <<9314568F76096D49830FE73A943AC137>]/Prev 83319>> A relation is any set of ordered pairs. Consider the relations consisting of the seven ordered pair solutions to \(y = |x| 2 \)and \(x = |y| + 1\). 39. f(x) = 3x2 5x; f(2) Answer 40. g(x) = 4x2 3x; g(3) 41. 4.2 Relations and Functions A function is a relation in . (a) True (b) False (c) True (d) True (e) False, 9. [ ] Restart your browser. It is a relation.) -2 Relation. Value of [f(0) + f(1)] / 2 is -2, (a) x = 3/2 (b) x = 3 (c) x = , 13. 5 0 obj 9. Given the graph of \(g(x)\), find \(g(8), g(0)\), and \(g(8)\). The relation is a function. The first 4 sheets each give:a quadratic equation a partially complete storya space for students to fill in the vertex, y-intercept, and x-intercept a table of values to fill ina space for students to write and answer their own question about the relationThe 5th sheet is a cha, Identifying Functions Independent Practice: Given 25 relations (ordered pairs, graph, table, mapping and equation), students will identify those relations which are NOT A FUNCTION by highlighting the boxes. Find \(f (4), f (0)\), and \(f (2)\).\. Introduction to New Material. Legal. If it is a function, give the domain and range. The value of a is 0.90 and b is 24.5, 11. /Title ( I n f i n i t e A l g e b r a 1 - C o n t i n u o u s R e l a t i o n s) x{|7>Z]-Ed]-lJVe;qN8v%vb;vnv$@ \CH(w Domain. Feel free to download and enjoy these free worksheets on functions and relations .Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Mathematics Homework Helper. 'y`u#$u:x7K#^o-6o%z}'It DOI^=--g@BB*V]& \>l w)y(K$D Zs&XI g"Dv+zZN7Oo,\u4BS\eYK.}%D\pV)UO^>-W3ud>JS%YE4n:P?ta\`UR;2M[!m?f4z+ao6>}khBqjY]~\ X+>XZC;I!4j|um=#b5>v5{umK)V+$6 4.0,` 3p H.Hi@A> Is the Equation a Function? endobj %PDF-1.3 % >> We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. /Resources 13 0 R \(g (2) = 4, g (1) = 3, g (0) = 4\), 9. <>>> The set of \(x\)-values defines the domain and the set of \(y\)-values defines the range. x ! There is exactly one output for every input. 5Pairs \((x, y)\) that identify position relative to the origin on a rectangular coordinate plane. If the function is f (x)= 6x-27, and the domain is {-5, 3, 15, 17}, then find the range. 11The set consisting of all of the first components of a relation. With the definition of a function comes special notation. Substitute \(f (x)\) with \(27\) and solve. The value of a certain automobile in dollars depends on the number of years since it was purchased in \(1970\) according to the following function: 11. 13 0 obj What will be the domain and range of the following relation? xXnF}-p(p#YBc),Iiy[ 9^:{wqs k8trv-@, Ls?[^~{;%_&d~tfn>C8Mg*d!?M'WHiRK w A! Who is credited with the introduction of the notation \(y = f (x)\)? Given the graph, state the domain and range and determine whether or not it represents a function: From the graph we can see that the minimum \(x\)-value is \(1\) and the maximum \(x\)-value is \(5\). The representation of a relation on a rectangular coordinate plane, as illustrated above, is called a graph10. 16The notation \(f (x) = y\), which reads \(f\) of \(x\) is equal to \(y\). Given a function, \(y\) and \(f (x)\) can be used interchangeably. Rational Numbers Between Two Rational Numbers, XXXVII Roman Numeral - Conversion, Rules, Uses, and FAQs, Find Best Teacher for Online Tuition on Vedantu. Range = {4, 16, 25, 36} and Domain = {-2, 2, 4, 5, 6}, 10. Students will determine if the given relations (shown in mapping diagrams, function tables, graphs, and pairs of points) are a function or not. Examples of domain and range, vertical line test, and determining if a relation is a function. This compilation of domain and range worksheet pdfs provides 8th grade and high school students with ample practice in determining the domain or the set of possible input values (x) and range, the resultant or output values (y) using a variety of exercises with ordered pairs presented on graphs and in table format. As we have seen, functions are also expressed using graphs. Yes 3. Explain to a beginning algebra student what the vertical line test is and why it works. %PDF-1.4 Domain: \(\); range: \([2, )\); function: yes, 15. Worksheet 7.4 inverse functions answer key - Download worksheet 7.4 inverse functions and more Elementary Mathematics Lecture notes in PDF only on Docsity! noncommercial purposes and are not distributed outside of a specific teacher's classroom. TPT empowers educators to teach at their best. Determine whether this relation is a function or not, and find the values of both a and b. Also sketch a graph for each equation. Section 3.1. Just print + go! 1.1 Functions and Function Notation - Precalculus 2e | OpenStax Uh-oh, there's been a glitch We're not quite sure what went wrong. (5, 4), (8, -1), (7, 3), (0, 5), (10, -2), and __________ << 1) 2x + 3y = 12 x-intercepts (let y = 0). /XObject << Algebra 2 relations and functions worksheet answers can be a useful tool for these scholars. Ordered Pair, Domain, Range, Function Notation, Composite Functions, Real-Life Functions.This 8 page Functions and Relations worksheet (with complete answer key) covers all . The students look at graphs containing dots and decide rather or not the dots form a linear or nonlinear relationship. The horizontal number line is called the x-axis2, and the vertical number line is called the y-axis3. Are you looking for a comprehensive set of worksheets to introduce functions to your students? \(\begin{array} { c } { f ( x ) = 5 x + 7 }\\\color{Cerulean}{\downarrow}\quad\quad\quad\:\:\: \\ { 27 = 5 x + 7 } \\ { 20 = 5 x } \\ { 4 = x } \end{array}\). Now includes an additional version that uses "y=" notation instead of "f(x)=" notation.This product is part of the Discovery-Based Worksheet Series. E6S2)212 "l+&Y4P%\%g|eTI (L 0_&l2E 9r9h xgIbifSb1+MxL0oE%YmhYh~S=zU&AYl/ $ZU m@O l^'lsk.+7o9V;?#I3eEKDd9i,UQ h6'~khu_ }9PIo= C#$n?z}[1 Given the graph of \(h\), find \(x\) where \(h(x)=-4\). How to Calculate the Percentage of Marks? /Font << A Explanations 1. \(g (10) = 10\) and \(g (5) = 10 ; g (5) = 5\) and \(g (10) = 5\). WORKSHEET 7.4 INVERSE FUNCTIONS Inverse Relations Find the inverse for each the inverse for each relation below (put your answer on the same graph). Relations and Functions Worksheet with Answer Key (PDF), Scatter Plot and Line of Best Fit Worksheet (PDF), Absolute Value Equations and Inequalities Worksheet (PDF), 2nd Grade Measuring Worksheet (with Answer Key), Square Numbers Worksheet (with Answer Key), Expanded Form Worksheet (with Answer Key). /Contents 11 0 R 2. . Avg. (c) All functions are relations, but all relations are not functions. Domain: \([5, 1]\); range: \([2, 2]\); function: no, 25. A relation with this property is called a function14. \(g ( 0 ) = 2 \sqrt { 2 } , g ( - 8 ) = 0 , g \left( a ^ { 2 } - 8 \right) = | a |\), 19. Domain: \((, 0]\); range: \([1, )\); function: yes, 19. 6The point where the \(x\)- and \(y\)-axes cross, denoted by \((0, 0)\). /Pages 3 0 R \(\begin{array} { c } { h ( \color{Cerulean}{- 2}\color{Black}{ )} = \frac { 1 } { 2 } ( \color{Cerulean}{- 2}\color{Black}{ )} - 3 = - 1 - 3 = - 4 } \\ { h ( \color{Cerulean}{0}\color{Black}{ )} = \frac { 1 } { 2 } ( \color{Cerulean}{0}\color{Black}{ )} - 3 = 0 - 3 = - 3 } \\ { h ( \color{Cerulean}{7}\color{Black}{ )} = \frac { 1 } { 2 } ( \color{Cerulean}{7}\color{Black}{ )} - 3 = \frac { 7 } { 2 } - 3 = \frac { 1 } { 2 } } \end{array}\). 0000001084 00000 n 26 13 Share a link to a page that you think others may find useful. Students will identify function rules by, Relations, Functions, Domain and Range Task CardsThese 20 task cards cover the following objectives:1) Identify the domain and range of ordered pairs, tables, mappings, graphs, and equations.2) Determine whether a relation is a function given ordered pairs, tables, mappings, graphs, and equations.A recording worksheet is also included for students to write down their answers as they use the task cards. Free to use Step-by-step explanations for every solution Exclusive how-to animations Sc. Range: Function: o 2 t 3 3 (3, 3) -2) Relation . Find \(x\) where \(g (x) = 1, g (x) = 0\), and \(g (x) = 3\). 2. endobj (x) = (9x/5) + 32 and the inverse is a function also. >> Then, they will locate the box with the exercise number in it on the next page and follow the directions for it according to their answer. [ -c R!z"^Ow,c . We offer the fastest, most expert tutoring in the business. /GSa 4 0 R Our team is dedicated to providing the best possible service to our customers. We have grown leaps and bounds to be the best Online Tuition Website in India with immensely talented Vedantu Master Teachers, from the most reputed institutions. Given the graph of the function \(f\), find the function values. Find \(f (8), f (0)\), and \(f (8)\). you CAN do math! \(\begin{aligned} g ( \color{Cerulean}{- 2}\color{Black}{ )} & = ( \color{Cerulean}{- 2}\color{Black}{ )} ^ { 2 } = 4 \\ g ( \color{Cerulean}{\frac { 1 } { 2 }}\color{Black}{)} & = ( \color{Cerulean}{\frac { 1 } { 2 }} \color{Black}{)} ^ { 2 } = \frac { 1 } { 4 } \\ g (\color{Cerulean}{ x + h}\color{Black}{ )} & = ( \color{Cerulean}{x + h}\color{Black}{ )} ^ { 2 } = x ^ { 2 } + 2 x h + h ^ { 2 } \end{aligned}\), \(g (2) = 4,\: g ( \frac{1}{2} ) = \frac{1}{4} ,\: g (x + h) = x^{2} + 2xh + h^{2}\), At this point, it is important to note that, in general, \(f (x + h) f (x) + f (h)\). Yes, each input has exactly 1 output. Lesson Directions. /Pattern << stream Here \(f\) is the function name, and \(f (x)\) denotes the value in the range associated with the value x in the domain. 0000047318 00000 n 8. You can get arithmetic support online by visiting websites such as Khan Academy or by downloading apps such as Photomath. Unfortunately, in the last year, adblock has now begun disabling almost all images from loading on our site, which has lead to mathwarehouse becoming unusable for adlbock users. 2?7Co?s[OC7>-o~?_/O>}w7$T']*Y)Q(~5s[e>Os~F}>5m&hJ1/W\sCJq/W\3HrI)rq375G9%%k7~O9i5e~~Ys3fn?*7VYsnOS37fhmj)4R*7a~W.O|a-F1gs/9%Vy>5DD{d-J?=95:'>)S}wxBsf/O|S|ftn~}Qh/fq(4M~BsZm %o(5nU/ 10/10 gives correct answers every time. Section 5.2 Defining Relations and Functions. trailer /CSp /DeviceRGB So, this relation is a function as well. The minimum \(y\)-value represented on the graph is \(0\); thus, the range is \([0,)\). With this data, a student maps a relationship between forehand and hand in the form y = ax + b, provided both are constants. hs2z\nLA"Sdr%,lt These two number lines define a flat surface called a plane4, and each point on this plane is associated with an ordered pair5 of real numbers \((x, y)\). /Length 12 0 R Then looks at examples of the 5 types: a set of ordered pairsa table of valuesa mappinga discrete graphan equation. 26 0 obj <> endobj 0000019902 00000 n 120 students are studying in class 11 divided among four sections. % 33. g(x) = 2x + 1 Answer 34. g(x) = 5x 8 35. g(x) = 3x 2 Answer 36. g(x) = 8x + 2 37. g(x) = 3 x Answer 38. g(x) = 7 5x In the following exercises, evaluate the function. FV>2 u/_$\BCv< 5]s.,4&yUx~xw-bEDCHGKwFGEGME{EEKX,YFZ ={$vrK Of special interest are relations where every \(x\)-value corresponds to exactly one \(y\)-value. On the other hand, an undefined function is the one whose points are outside the domain. \(\begin{aligned} f ( \color{Cerulean}{- 2}\color{Black}{ )} & = \sqrt { 2 ( \color{Cerulean}{- 2}\color{Black}{ )} + 4 } = \sqrt { - 4 + 4 } = \sqrt { 0 } = 0 \\ f ( \color{Cerulean}{0}\color{Black}{ )} & = \sqrt { 2 ( \color{Cerulean}{0}\color{Black}{ )} + 4 } = \sqrt { 0 + 4 } = \sqrt { 4 } = 2 \\ f ( \color{Cerulean}{\frac { 1 } { 2 } a ^ { 2 } - 2}\color{Black}{ )} & = \sqrt { 2( \color{Cerulean}{ \frac { 1 } { 2 } a ^ { 2 } - 2}\color{Black}{)} + 4 } = \sqrt { a ^ { 2 } - 4 + 4 } = \sqrt { a ^ { 2 } } = | a | \end{aligned}\), \(f (2) = 0,\: f (0) = 2,\: f \left( \frac { 1 } { 2 } a ^ { 2 } - 2 \right)= |a|\). Questions 4-12 present students with relations in any of the given formats and ask to identify the domain, range, determine whether the relation is a function, and provide an explanation.An answer key is provided!This resource is also i, This ready to use product is a quick, fun way to have your students practice identifying functions. Worksheet 4.1 Relations and Functions Write each of the following as a relation, state the domain and range, then determine if it is a function. Answers vary, should mention how the function does not always have the same output for a given input. [7A\SwBOK/X/_Q>QG[ `Aaac#*Z;8cq>[&IIMST`kh&45YYF9=X_,,S-,Y)YXmk]c}jc-v};]N"&1=xtv(}'{'IY) -rqr.d._xpUZMvm=+KG^WWbj>:>>>v}/avO8 If you're looking for a quick delivery, we've got you covered. It is important to note that \(y\) and \(f(x)\) are used interchangeably. Analyzing Relations and Functions KEY. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Is the relation a function? %PDF-1.5 Function: Domain: 10 Range: Function. The zip file contains the worksheet in both .doc format as well as in .pdf. \(f ( - 2 ) = - 7 , f ( 0 ) = - 3 , f ( x - 3 ) = 2 x - 9\), 7. This is a relations and functions worksheet. A function is a relation in which each element of the domain is paired with EXACTLY one element of the range. x y3 -4-52-7 -4-5 7A function is a relation in which each input has exactly one outputThe domain of the function is the set of all input values. LCM of 3 and 4, and How to Find Least Common Multiple, What is Simple Interest? Function. I suggest using this as a note guide during the lesson. 25. Plus each one comes with an answer key. Step 1: Tell the class a story involving a real-world, linear functional relationship. << >> 0000001728 00000 n <> Domain: \(\); range: \((, 3]\); function: yes, 21. If asked to find \(x\) where \(f(x) = a\), we set the function equal to \(a\) and then solve for \(x\). Great resource to use as homework or independent work.Answer Key provided.You may also be interested in these Functions resources:Evaluating Functions Coloring ActivityIdentifying Functions Review WorksheetIdentifying Functions Card Sort ActivityFunctions: Multiple Representations Matching A, This is a one-sided fill-in-the-blank notes page on differentiating between a function and a relation, exploring the 5 types of functions/relations, and then finding the domain and range of each. Find \(x\) where \(g (x) = 3, g (x) = 0\), and \(g (x) = 2\). These worksheets cover all the essential concepts related to functions, including domain and range, linear versus nonlinear, graphing stories, and much more. . NhSIS+:|2q^>l$ia}^nCLW:'HdfJ)A3X3&X "80j]./j\^4z It is NOT a function since two ordered pairs have the same domain or x-value. Find the value of x that will make this relation a function. 4. 7. Evaluate the inverse and also rule out whether the same is a function or not. Posted: 7/3/17 so 50% off through 7/6/17, Do your students need practice graphing real world quadratics? As we have seen, functions are also expressed using graphs. 38 0 obj <>stream /F9 9 0 R Plus each one comes with an answer key. Relation. 2. What was the value of the car new? Is the relation a function? Domain: \(\); range: \([0, ]\); function: yes, 27. P x Q provides number of relations to be 64. You must try to solve the questions on your own and later check it with the given practice worksheet relations and functions answer key. Range. Explain your reasoning for each. Explain your reasoning. If so, state the domain and range. Holt Algebra 4.2 Relations and Functions PPT + Worksheet. endstream endobj 27 0 obj <> endobj 28 0 obj <> endobj 29 0 obj <>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]/XObject<>>> endobj 30 0 obj <> endobj 31 0 obj [/ICCBased 34 0 R] endobj 32 0 obj <> endobj 33 0 obj <>stream With a wide variety of practice problems, your students will gain a deep understanding of these fundamental concepts and. Domain: {3, 6, 9, 12}; Range: {-4, 0, 4} Mapping Diagrams: given a mapping diagram of a relation, we can tell if it is a function if each input is paired with exactly 1 output. Functions Worksheet 1 For exercises 13-18, determine whether each relation is a function. Set P = {-2,0,2} and the remaining elements are { (-2,-2), (-2,2), (0,-2), (0,0), (2,-2), (2,0), (2,2)} 2. An expression is considered to be unambiguous or well-defined when the definition of the same allocates a unique value or interpretation. In what years was the car valued at \($4,000\)? There's more to your application than just filling out the forms. Worksheet #1 Determine whether each relation is a function or not. In mathematics, a student will get to study multiple kinds of functions, which are one one, many one, onto, into, polynomial, linear, identical, quadratic, rational, cubic, algebraic, modulus, signum, greatest integer, fractional part, even and odd, periodic, composite, constant, identity, etc. Use the function to determine the salespersons income if he sells \(3\) cars this month. /Filter /FlateDecode The domain is \(\{4, 2, 0, 3\}\) and the range is \(\{3, 3, 5, 6, 7\}\). Then graph the function and its inverse. Chapter 1 - Analyzing Functions Answer Key CK-12 Math Analysis Concepts 1 1.1 Relations and Functions Answers 1. . [/Pattern /DeviceRGB] The notation \(f (x)\) is read . 4The flat surface defined by \(x\)- and \(y\)-axes. a. b. Chapter 3: Introduction to Functions and Relations. The \(x\)- and \(y\)-axes break the plane into four regions called quadrants7, named using roman numerals I, II, III, and IV, as pictured. If any of the x values are repeated, but the values that correspond to them in the y-axis are different, then we are dealing with a relation rather than a function. Domain and Range Worksheets. 5. Very interesting and supportful Thank you , haven't gotten one guest one wrong yet, really good app for homework. Reflections Over Intersecting Lines as Rotations. This download also includes a PDF "worksheet" with the exact same questions as the Google Form. Illustrate the travelled distance in the form of function t in hours. { "201:_Relations_Graphs_and_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "202:_Linear_Functions_and_Their_Graphs" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "203:_Modeling_Linear_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "204:_Graphing_the_Basic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "205:_Using_Transformations_to_Graph_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "206:_Solving_Absolute_Value_Equations_and_Inequalities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "207:_Solving_Inequalities_with_Two_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "20E:_2E:_Graphing_Functions_and_Inequalities_(Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Algebra_Fundamentals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Graphing_Functions_and_Inequalities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Solving_Linear_Systems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Polynomial_and_Rational_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Radical_Functions_and_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Solving_Equations_and_Inequalities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Exponential_and_Logarithmic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Conic_Sections" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Sequences_Series_and_the_Binomial_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "license:ccbyncsa", "showtoc:no", "authorname:anonymous", "licenseversion:30", "program:hidden", "source@https://2012books.lardbucket.org/books/advanced-algebra/index.html" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FAlgebra%2FBook%253A_Advanced_Algebra%2F02%253A_Graphing_Functions_and_Inequalities%2F201%253A_Relations_Graphs_and_Functions, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), source@https://2012books.lardbucket.org/books/advanced-algebra/index.html, status page at https://status.libretexts.org.

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