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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].
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.
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
School Science and Mathematics, 1992
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.
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.
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).
2013
The domain of a function is the set of x values (along the x-axis) that gives a valid answer (y value) when the function is evaluated. Also, the set of all x values must be mapped to one and only one y value.
1988
This report summarizes the work of a two-year project which focused primarily on the problems that students have with algebra in general, and graphs in particular. The first of two major sections in the document deals with the use of computer software to assist in the teaching of graphing. It concludes that thoughtful design and use of graphing software presents new opportunities for teaching about graphing. The next section of the report centers on the development of research instruments that are intended to study scale in the context of graphs of function. It includes a set of problem-based teaching materials that were used as research tools. The appendices contain descriptions of probes designed to see if students can interpret and create graphs of real-world phenomena, along with instruments dealing with mapping, scale, and computer explorations. (TW)
linear and quadratic functions a brief description
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 ...
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