Functions and graphs
Lawrence As
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Abstract
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This chapter provides a comprehensive exploration of functions, including definitions, domains, ranges, and various types of functions. It also discusses the graphical representation of functions, inequalities involving polynomials, and the continuity of functions based on their graphs.
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Functions and Their Graphs REPRESENTING RELATIONS AND FUNCTIONS
A is a mapping, or pairing, of input values with output values. The set of input values is the and the set of output values is the A relation is a provided there is exactly one output for each input. It is not a function if at least one input has more than one output.
Functions 1.1 Functions and Graphs
1. f (−5) = (−5) 2 − 1 = 25 − 1 = 24 f (− √ 3) = (− √ 3) 2 − 1 = 3 − 1 = 2 f (3) = (3) 2 − 1 = 9 − 1 = 8 f (6) = (6) 2 − 1 = 36 − 1 = 35 2. f (−5) = −2(−5) 2 + (−5) = −2(25) − 5 = −55 f (− 1 2) = −2(− 1 2) 2 + (− 1 2) = −2(1 4) − 1 2 = −1 f (2) = −2(2) 2 + (2) = −2(4) + 2 = −6 f (7) = −2(7) 2 + (7) = −2(49) + 7 = −91 3. f (−1) = √ −1 + 1 = √ 0 = 0 f (0) = √ 0 + 1 = √ 1 = 1 f (3) = √ 3 + 1 = √ 4 = 2 f (5) = √ 5 + 1 = √ 6 4. f (− 1 2) = ï¿¿ 2(− 1 2) + 4 = √ −1 + 4 = √ 3 f (1 2) = ï¿¿ 2(1 2) + 4 = √ 1 + 4 = √ 5 f (5 2) = ï¿¿ 2(5 2) + 4 = √ 5 + 4 = √ 9 = 3 f (4) = ï¿¿ 2(4) + 4 = √ 8 + 4 = √ 12 = 2 √ 3 2
Concept of function
International Journal of Scientific & Engineering Research,, 2017
This study presents investigation of difficulties of grade nine students have on the concept of function using a descriptive survey method. Both quantitative and qualitative research approaches were used to explore the students’ challenges in this study. Data was collected using test, interview and teachers’ questionnaire. The four commonly used ways to represent function is then used to analyse students’ interpretation and manipulation in different contexts. The finding of the study reviled that students have difficulties in basic definition of function, to identify the difference between function and relation, difficulty to identify an equation with two variables x and y as a function or not. The students have also difficulty in verbal and graphical representation of function. Identified students misunderstanding about the function concept are: any functions should contain both x and y variables, any continuous graph are a function and a discontinuous graph is not a function. This could be because of students’ attention to the lesson, their background weakness and students limited English language skills. Based on the findings of the study different recommendations are suggested in order to solve the problems.
Function and Graph in DGS Environment
International Group for the Psychology of Mathematics Education, 2005
Since a long time, the notion of function has been at the core of a great number of studies, and the rich literature reports on a number of difficulties related to different aspects of the notion of function (Goldenberg, Lewis and O’Keefe 1992, Harel and Dubinsky 1991, Sfard 1991, Sierpinska 1992, Tall 1991, Vinner and Dreyfus 1989, Leinhardt, Zaslavky & Stein 1990). Difficulties of interpreting graphic information in terms of function are widely reported. It seems that for students there is a lack of explicit relationship between function and graph1, (Vinner & Dreyfus, 1989, Dreyfus & Eisenberg, 1983); students are not able to move from the one to the other identifying domain and image of function or confounding decreasing behavior with negative values (Trigueros, 1996). Rigid and stereotyped ideas are often related to functions and their graphs (Markovits et al. , 1986, 1988 ; Schwarz & Hershkowitz, 1996).
Basic functions and operations on functions
1989
We define the following mappings: the characteristic function of a subset of a set, the inclusion function (injection or embedding), the projections from a Cartesian product onto its arguments and diagonal function (inclusion of a set into its Cartesian square). Some operations on functions are also defined: the products of two functions (the complex function and the more general product-function), the function induced on power sets by the image and inverse-image. Some simple propositions related to the introduced notions are proved. MML Identifier: FUNCT_3.
Functions from a set to a set
Formalized Mathematics, 1990
Summary. The article is a continuation of [1]. We define the following concepts: a function from a set X into a set Y, denoted by Function of X, Y, the set of all functions from a set X into a set Y, denoted by Funcs (X, Y), and the permutation of a set (mode Permutation of X, ...
Graphing inequalities, connecting meaning
The Mathematics Teacher, 2014
The trichotomy property and order relation symbols show the Cartesian plane divided into three regions, as students connect tabular and graphical representations of functions.
How are graphs of two-variable functions taught?
This is a study about how graphs of functions of two-variables are taught. We are interested in particular in the techniques introduced to draw and analyze these graphs. This continues previous work dedicated to students' understanding of topics of twovariable functions in multivariable calculus courses. The model of the "moments of study" from the Anthropological Theory of the Didactic (ATD) is used to analyze the didactical organization of the topic of interest in a popular calculus textbook, and in a typical classroom presentation. In so doing we obtain information about the institutional dependence of findings in previous studies.
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Application of the Function in Basic Mathematics
International Journal of Advanced Science and Engineering, 2023
The concept of function is one of the fundamental mathematical concepts, very important within mathematics itself as well as in the application of mathematics. Functions are an essential element of mathematical structuring and modeling of problems (e.g.in algebraic structures), as well as a means of comparing structures thus obtained (eg homomorphisms of structures). A mathematical function is a rule that gives the value of the dependent variable corresponding to certain values of one or more independent variables. A function can be represented in several ways, such as a table, formula, or graph. Apart from isolated points, the mathematical functions found in physical chemistry are single-valued. Apart from isolated points, the mathematical functions that occur in physical chemistry are continuous.
FUNCTIONS AND MODELS 1.1 Four Ways to Represent a Function
In exercises requiring estimations or approximations, your answers may vary slightly from the answers given here. 1. (a) The point (−1, −2) is on the graph of f , so f (−1) = −2. (b) When x = 2, y is about 2.8, so f (2) ≈ 2.8. (c) f (x) = 2 is equivalent to y = 2. When y = 2, we have x = −3 and x = 1. (d) Reasonable estimates for x when y = 0 are x = −2.5 and x = 0.3. (e) The domain of f consists of all x-values on the graph of f. For this function, the domain is −3 ≤ x ≤ 3, or [−3, 3].
Examples and Non-examples as a Road to Understanding the Concept of Function
Eduma : Mathematics Education Learning and Teaching, 2020
The aim of this qualitative study is to describe the conceptual understanding of prospective mathematics teachers at STKIP PGRI Lumajang by determining examples and non-examples of functions. This research was conducted by giving tests and interviews to 12 subjects. Data were analyzed using Miles & Hubberman model. The results of this study showed that the subjects investigated whether paired to to determine a given statement is a function or not. Subjects focus on specific expressions, e.g. 2 or others that they have previously recognized as a function. However, some subjects used the vertical line test to determine a given graph represent function or not, although they were unable to explain why the vertical line test was the appropriate method. Their understanding is limited to procedural knowledge such as mental representations and incomplete concept images, then failing to be comprehensively linked to the definition of the concept of function.
Functions and their basic properties
Formalized Mathematics, 1990
FORMALIZED MATHEMATICS Number 1, January 1990 Université Catholique de Louvain Functions and Their Basic Properties Czes law Bylinski1 Warsaw University Bia lystok Summary. The definitions of the mode Function and the graph of a function are introduced. The graph of a function ...
THE FUNCTION CONCEPT: COMPREHENSION AND COMPLICATION
bsrlm.org.uk
In this paper, we contrast the mathematical simplicity of the function concept that is appreciated by some students and the spectrum of cognitive complications that most students have in coping with the function definition in its many representations. Our data is based on interviews with nine (17-year old) students selected as a crosssection from 114 responses to a questionnaire. We distinguish four categories in a spectrum from those who have a simple grasp of the core function concept applicable to the full range of representations to those who see only complicated details in different contexts without any overall grasp of the conceptual structure.
Function Concept and Transformations of Functions: The Role of the Graphic Calculator
2011
This communication focuses on the activity of two secondary school students in a task involving the concept of function and transformations of functions of the type () p x k and () p x h , and analyses the role of the graphic calculator. The results suggest that students are beginning to have an object-oriented view of function. Concerning the transformation () p x k they have already established the operational invariants, and it seems that the graphic calculator played an important role in that.
Graphs of functions of two variables: results from the design of instruction
International Journal of Mathematical Education in Science and Technology, 2013
In a series of previous studies, the authors have described specific mental constructions that students need to develop, and which help explain widely observed difficulties in their graphical analysis of functions of two variables. This new study, which applies Action-Process-Object-Schema theory and Semiotic Representation Theory, is based on semi-structured interviews with 15 students. It results in new observations on student graphical understanding of two-variable functions. The effect of research findings in designing a set of activities to help students carry out the specific constructions found to be needed is briefly discussed.
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