...Composition and Inverse Alicia Frambro MAT 222: Intermediate Algebra Prof: Michael Smith September 22, 2013 Composition and Inverse When using functions, there are different ways to solve various values. Many companies use these functions to monitor business pertaining gross profit and many other operations such as adjusting productivity. Visual examples of these functions are graphs. They provide a visual relationship between composition and inverse solutions as well and the difference between profit gain and profit loss for a business. The following functions will be used to solve certain problems. fx=2x+5 g(x)=x2-3 (x) = 7+x3 f-h(4) Use rule of composition and solve which is f(4)-h(4). f(x)-h(x)= 2x+5-(7-x) 2(4) +5- 7-43 Substitute 4 for x. 8+ 5- 33 Use order of operations to solve. 8+5-1= 12 Therefore (f-h)(4)= 12 Evaluate two compositions of the above functions. A. (fog)(x) B. (fog)(x) The method that should be used to solve is to find the solution of one function and then substitute the solution for x in the other solution. For A we would first solve for g(x) and substitute the solution for x into f(x). A. (fog)(x)= f(g(x)) (fog)(x)= f(x2-3) (fog)(x)= 2(x2-3) +5 Substitute g(x) for x in f(x). (fog)(x) = 2x2-6+5 Simplify by using the distributive property and order of operations. (fog)(x)= 2x2 -1 Answer. B. (h0g)(x) = h(g(x)) Use the same concept of A but (hg)(x)= h(g(x)) (hog)(x) = h(x2 – 3) ...
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...Composition and Inverse Functions provide an opportunity for manipulating expressions using different values. These values can help business owners, data analysts, and even the consumer compare rates and data. Functions also extend independent (x) and dependent (y) variables by graphing in the coordinate plane and creating a visual demonstration of the relationship. The following functions will be used in the required problems. f(x) = 2x+5 g(x) = x2+3 h(x) = (7-x)/3 The first task is to compute (f – h)(4). (f – h)(4) = f(4) – h(4) Because of rules of composition, each function may be calculated separately and then subtracted. f(4) = 2(4) + 5 The ‘x’ was replaced with the 4 from the problem. f(4) = 8 + 5 Order of operations was used to evaluate the function. f(4) = 13 h(4) = (7 - 4)/3 The same process is used for h(4) and f(4). h(4) = 3/3 h(4) = 1 (f – h)(4) = 13 – 1 (f – h)(4) = 12 This is the solution after plugging the values in and subtracting. Next, two pairs of the functions will be composed into each other. One option to find the solution for the function, g(x), is to calculate it and then substitute for the ‘x’ value in the f(x). The option used here is to replace the ‘x’ in the ‘f’ function with the ‘g’ function. This means the rule of ‘f’ will work on ‘g’. This means the rule of ‘f’ will work on the problem: (f°g)(x) = f(g(x)) (f° g)(x) = f(g(x)) (f° g)(x) = f(x2 + 3) ‘f’ is now going to work on the rule...
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...Composition and Inverse MAT222: Week 5 Assignment April 6, 2014 Composition and Inverse For this week’s assignment we are given the task of learning how to solve Composition and Inverse math problems. This week’s assignment focuses on the following problem: f(x)=2x+5 g(x)=x2-3 h(x)= 7-x / 3 The first thing I will do is compute (f – h)(4). (f – h)(4)=f(4) – h(4). Each function may be calculated separately and will be subtracted due to the rules of composition. f(4)=2(4)+5 We will then substitute the 4 from the problem and plug it into the x. f(4)=8 +5 We will be using order of operations in order to evaluate the function. f(4)=13 h(4)=(7-4) / 3 The same process will be used in this function where we will plug in f(4) and h(4) then the problem will look like: h(4)=3/3 h(4) = 1 (f – h)(4)=13-1 (f – h)(4)=12 This is the solution after substituting the values into the problem. The next step will consist of the two pairs of functions will composed into each other. In order to do this I will first have to find the solution for the function g(x). In order to do this I will be calculating it and then substituting for the x value in the f(x). This rule will function because the g function will be replacing the f function. Therefore this rule will help us (f▫g)(x)= f(g(x)). (f▫g)(x)=f(g(x)) (f▫g)(x)=f(x2-3) f will work on the rule of g and g will be replacing x. (f▫g)(x)=2(x2 - 3)-5 we will be using the rule of f and it will be applied...
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...Composition and Inverse Jessica Cantu MAT 222 Week5 Final Assignment John Gomillion June 13, 2014 Composition and Inverse Functions can be very useful because they give us the option to change up an expression by using different values. These values and functions can help a person or a company be successful and wise. Functions also extend independent (x) and dependent (y) variables by graphing in the coordinate plane. This is very helpful because it creates a visual demonstration of the relationship. The following functions will be used in the required problems. f(x) = 2x + 5 g(x) = x2 – 3 h(x) = 7 – x 3 The first task is to compute (f – h)(4). (f – h)(4) = f(4) – h(4) Following the rules of composition, each function can be calculated separately and then subtracted. f(4) = 2(4) + 5 The x is replaced with the 4 from the first problem. f(4) = 8 + 5 Order of operations is used to evaluate the function. f(4) = 13 h(4) = (7 – 4)/3 Here the same process is used for h(4) and f(4). h(4) = 3/3 h(4) = 1 (f – h)(4) = 13 – 1 (f – h)(4) = 12 This is the solution after substituting...
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...After the previous difficulties absorbing all of the required material during unit three, I was slightly apprehensive going into unit four. Especially given that the awaiting concepts had such monikers as function composition and inverse functions do not aid in easing my fears. However, to my surprise, my fears were ill-founded as I discovered that I comprehended the material for this unit easier and quicker than I have for any unit learned so far. This of course does not imply that I am under the belief that I am now only a few steps away from mastery as much as it means that progress is being made and there is hope that I can grow in my ability to solve problems of a mathematical nature. The authors of the textbook explain function compositions and its related concepts thoroughly and concisely. Subsequently, I only required two readings to feel confident in my comprehension of the material. Remembering that the function composition (f o g) is equivalent to the statement f(g(x)) simplified the learning process for the rest of the first chapter. Furthermore, as the left to right arrangement of the functions is equivalent for both (f o g) and f(g(x)), committing this fact to memory was not overtly difficult....
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...Composition and Inverse Tara Fuentes Mat 222 Week 5 Assignment Kristopher Childs 1/19/15 In this week's assignment we need to complete composite and inverse functions. The x and y interchange when a function is inversed, otherwise the points are identical. This is the following function. f(x)=2x+5 g(x)=x2-3 h(x)= 7-x/3 First we need to compute (f-h)(4) (f*h)(4)=f(4)-h(4), each function can be done separately f(4)=2(4)+5 f(4)=8+5 f(4)=13 H h(4)=(7-4)/3 same process as above h(4)=3/3=h(4)=1 (f-h)(4)=13-1 (f-h)(4)=12 this is the solution after substituting and subtracting The next part we need to replace the x in the f function with the g (f*g)(x)=f(g(x)) (f*g)(x)=f(x2-3) (f*g)(x)=2x2-1 is the result Now we need to do the h function (h*g)(x)=h(g(x)) (h*g)(x)=h(x2-3) (h*g)(x)=7-(x2-3) (h*g)(x)=10-x2 end result The inverse function-- f-1(x)=x-5h-1(x)=-(3-7) By doing problems this way it can save a person and a business a lot of time. A lot of people think they don't need math everyday throughout their life, but in all reality people use math almost everyday in life. The more you know the better off your life will...
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...1. Read Module 3 Topic 7, Module 4 Topic 1 and 2, Module 5 Topic 1~2. 2. Do the drills for the topics. 3. Read the Chapter 3 sections 2, 5 and Chapter 4 sections 1~3 in your textbook. 4. Do Homework for week 5 (you can find the list in the conference). Week 5 Supplementary Notes Chapter 3 Section 3.2: Polynomial Function of Higher Degree A polynomial function P is given by , where the coefficients are real numbers and the exponents are whole numbers. This polynomial is of nth degree. Far-Left and Far-Right Behavior The behavior of the graph of a polynomial function as x becomes very large or very small is referred to as the end behavior of the graph. The leading term of a polynomial function determines its end behavior. x becomes very large x → ∞ x becomes very large x → ∞ x becomes very small -∞ ← x x becomes very small -∞ ← x We can summarize the end behavior as follows: The Leading-Term Test If is the leading term of a polynomial, then the behavior of the graph as x → ∞ or as x → −∞ can be described in one of the four following ways. If n is even and an >0: ▼ ▼ | If n is even and an <0:▲ ▲ | If n is odd and an >0: ▲▼ | If n is odd and an <0: ▲ ▼ | Polynomial Function, Real Zeros, Graphs, and Factors (x − c) If c is a real zero of a function (that is, f(c)=0), then (c,0) is an x-intercept of the graph...
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...applications using linear systems as well as evaluating and finding partial sums of a series. Course Objectives After completing this course, students will be able to: ● Identify and then calculate perimeter, area, surface area, and volume for standard geometric figures ● Perform operations on real numbers and polynomials. ● Simplify algebraic, rational, and radical expressions. ● Solve both linear and quadratic equations and inequalities. ● Solve word problems involving linear and quadratic equations and inequalities. ● Solve polynomial, rational, and radical equations and applications. ● Solve and graph linear, quadratic, absolute value, and piecewise-defined functions. ● Perform operations with functions as well as find composition and inverse functions. ● Graph quadratic, square root, cubic, and cube root functions. ● Graph and find zeroes of polynomial...
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...Historical note: Newton used y, while Leibniz used dx . About a century later Lagrange introduced y and ˙ Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below. 4. Operating Principle Many functions are formed by successively combining simple functions, using constructions such as sum, product and composition. To differentiate, apply the differentiation rule corresponding to the last construction. 5. Rules for Constructions SUM: LINEARITY: PRODUCT: RECIPROCAL: QUOTIENT: CHAIN (for compositions): D( f + g) = D f + Dg D(a f + bg) = aD f + bDg D( f · g) = D f · g + f · Dg D(1/ f ) = −D f / f 2 g · D f − f · Dg g2 D( f ◦ g) = (D f ) ◦ g · Dg dy dy du = dx du dx D( f /g) = 1...
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...Science of Stars This paper is about the science of the stars. In this paper I will address how astronomers determine the composition, temperature, speed, and rotation rate of distant objects. I will briefly explain the properties of stars in the H-R diagram from Chapter 15 of the course textbook “The Cosmic Perspective”. I will also summarize the lifecycle of the Sun and identify where the Sun is in its lifecycle. Studying Distant Objects Astronomers study light which comes from distant objects to determine its composition, temperature, speed, and rotation of distant objects. This process is called spectroscopy. Spectroscopy was first used to study celestial objects in 1863 by William Higgins. By using this process he discovered the Sun and most stars are primarily composed of hydrogen gases. By using the spectroscopy technique it was discovered that different objects give off and absorb different spectrums of light. Where the object falls in the spectrum of light can be determined by examining its peak intensity at each wave length of light. The light helps us to determine an objects composition, temperature, and rotation. There are three types of spectra used to evaluate light. Objects which absorb light at different wavelengths are referred to as absorption spectrum. The intensity of light drops in objects which absorb light and therefore appear as dark lines on a rainbow of colors. Objects such are stars, planets with atmospheres, and galaxies absorb light...
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...+mohammad بحث صور خرائط Youtube الأخبار Gmail Drive المزيد full_stop_900@hotmail.com 0 مشاركة… mohammad alyahya mohammad alyahya malyahya0001@stu.kau.edu.sa Google البحث الآمن ▼ حوالي ٣٤٬٢٠٠ من النتائج (عدد الثواني: 0,48) نتائج البحث Handbook of Automated Essay Evaluation: Current ... - صفحة 125 https://books.google.com.sa/books?isbn... - ترجم هذه الصفحة Mark D. Shermis, Jill Burstein - 2013 - معاينة - المزيد من الإصدارات Current Applications and New Directions Mark D. Shermis, Jill Burstein. models resulting from machine learning must be effective at reproducing human assessment, which requires weighing the complexity and thoroughness of a ... Machine Learning Algorithms for Problem Solving in ... - صفحة 136 https://books.google.com.sa/books?isbn... - ترجم هذه الصفحة Kulkarni, Siddhivinayak - 2012 - معاينة - المزيد من الإصدارات It consists of essays written by English language students who are studying English in their third or fourth year at university. The corpus currently has over 3 million words from students from 16 different native languages. The target for each ... Psychology of Learning and Motivation: Advances in ... - صفحة 58 https://books.google.com.sa/books?isbn... - ترجم هذه الصفحة Brian H. Ross - 2002 - معاينة - المزيد من الإصدارات The larger the number and variety of essay grades there were to mimic, the...
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...During the first six months, there were four cooking classes and four follow up sessions offered (Mellberg et al. 2014). At the beginning of the study, anthropometric measurements were taken and then later taken at 6, 12, 18, and 24 months. These measurements included abdominal diameter, body composition, blood pressure, glucose levels, and energy expenditure. Participants were asked to keep a food record for four days (which included 3 weekdays and 1 weekend day). The...
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...heat and/or acid and in that way they remain intact during the whole production of the food till the consumption (Sicherer and Sampson 2006). As it was described before, food-induced allergic disorder is an Immunoglobulin E (IgE) -mediated hypersensitivity reaction. During the initial intake of the food, the allergenic protein stimulates the production of IgE antibodies which bind to tissue basophils and mast cells and trigger the release of mediators (histamine, prostaglandins and leukotrienes) causing allergenic symptoms (Sicherer and Sampson 2010). There is not specific treatment for food allergies. However, a proper management and well-balanced diet and avoidance of highly allergenic food in mothers and infant diets (can reduce the inverse side-effects of these food allergens (Waserman and Watson 2011). CMA is the most common during the infancy with a prevalence of 3% of the general child population (Hudson 1995). Cow’s milk allergy can be treated only with completely elimination from infant’s diet of the responsible antigen showing alternation of the symptoms (Isolauri 1997). Breastfeeding of the babies is considered the golden standard for infant nutrition, the development of hypoallergenic infant formulas are a good replacement of human’s milk for the babies diagnosed with CMA of at risk of developing it (Knipping, Simons et al. 2014). Hypoallergenic infant formulas are whey protein partially or extensively hydrolyzed products derived by enzymatic, heat and/or ultrafiltration...
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...Abscissa - word for the x part of a coordinate pair. Absolute value - The positive value of the indicated number or expression./an operation that tells you how far a number is from zero. Additive inverse - Number with the same numerical part but the opposite sign (plus or minus) of the given number. If zero is the sum of two numbers, then these two numbers are additive inverses of one another Area - the amount of space covered by a two-dimensional object./measure of a specified region in a plane. Associative property - characteristic of addition and multiplication that allows the grouping of terms to change without affecting the result. Asymptote - Boundary line that a graph gets infinitely close to but never actually touches. Base - in the expression x to the second power, the base is x; x will be multiplied by itself two times./ value multiplied repeatedly in an exponential expression. Binary operation - process requiring two values to produce a third value. Binominal - two terms separated by addition or subtraction. Bomb method - technique used to factor quadratic polynomials whose leading coefficient are not equal to 1; also called factoring by decomposition. Circumference - distance around the outside of a circle. Coefficient - number muliplied by a variable./ the number appearing at the beginning of a monomial; the coefficient of 12xy to the second power is 12. Combinations - method of counting that tells how many ways a designated number...
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...Contents 0. Preface 1. Functions and Models 1.1. Basic concepts of functions 1.2. Classification of functions 1.3. New functions from old functions 1 2 2 5 8 0. Preface Instructor: Jonathan WYLIE, mawylie@cityu.edu.hk Tutors: Radu Gogu, rgogu2@student.cityu.edu.hk. Texts: Single Variable Calculus, by James Stewart, 6E. In this semester, we will cover the majority of Chap 1-4, 7, 12. Upon completion of this course, you should be able to understand limit, derivatives, and its applications in mathematical modeling and infinite series. 1 2 1. Functions and Models In this chapter, we will briefly recall functions and its properties covered by high school. 1.1. Basic concepts of functions. Text Sec1.1: 5, 7, 39, 57, 67. Definition 1.1. A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. Usually, we write a function f : x → f (x) where (1) x ∈ D, i.e. x belongs to a set D , called the Domain; (2) f (x) ∈ E, i.e. f (x) belongs to a set E, called the Range; (3) x is independent variable, (4) f (x) is dependent variable. 3 For a function f , its graph is the set of points {(x, f (x)) : x ∈ D} in xy-plane. One can also use a table to represent a function. Example 1.1. Sketch the graph of following two piecewise defined functions. (1) f (x) = |x|. i.e. Absolute value of x. (2) f (x) = [x]. i.e. largest integer not greater than x. The graph of a function is a curve. But the question is: which curves are graphs of...
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