# Jackie Nicholas Janet Hunter Jacqui Hargreaves-Books Pdf

03 Feb 2020 | 34 views | 0 downloads | 78 Pages | 1,004.86 KB

Share Pdf : Jackie Nicholas Janet Hunter Jacqui Hargreaves

Download and Preview : Jackie Nicholas Janet Hunter Jacqui Hargreaves

Report CopyRight/DMCA Form For : Jackie Nicholas Janet Hunter Jacqui Hargreaves

## Transcription

Mathematics Learning Centre University of Sydney i. 1 Functions 1,1 1 What is a function 1,1 1 1 De nition of a function 1. 1 1 2 The Vertical Line Test 2,1 1 3 Domain of a function 2. 1 1 4 Range of a function 2, 1 2 Specifying or restricting the domain of a function 6. 1 3 The absolute value function 7,1 4 Exercises 8,2 More about functions 11. 2 1 Modifying functions by shifting 11,2 1 1 Vertical shift 11.
2 1 2 Horizontal shift 11,2 2 Modifying functions by stretching 12. 2 3 Modifying functions by re ections 13,2 3 1 Re ection in the x axis 13. 2 3 2 Re ection in the y axis 13,2 4 Other e ects 14. 2 5 Combining e ects 14,2 6 Graphing by addition of ordinates 16. 2 7 Using graphs to solve equations 17,2 8 Exercises 19.
2 9 Even and odd functions 21,2 10 Increasing and decreasing functions 23. 2 11 Exercises 24,3 Piecewise functions and solving inequalities 27. 3 1 Piecewise functions 27,3 1 1 Restricting the domain 27. 3 2 Exercises 29,3 3 Inequalities 32,3 4 Exercises 35. Mathematics Learning Centre University of Sydney ii. 4 Polynomials 36,4 1 Graphs of polynomials and their zeros 36.
4 1 1 Behaviour of polynomials when x is large 36,4 1 2 Polynomial equations and their roots 37. 4 1 3 Zeros of the quadratic polynomial 37,4 1 4 Zeros of cubic polynomials 39. 4 2 Polynomials of higher degree 41,4 3 Exercises 42. 4 4 Factorising polynomials 44,4 4 1 Dividing polynomials 44. 4 4 2 The Remainder Theorem 45,4 4 3 The Factor Theorem 46.
4 5 Exercises 49,5 Solutions to exercises 50, Mathematics Learning Centre University of Sydney 1. 1 Functions, In this Chapter we will cover various aspects of functions We will look at the de nition of. a function the domain and range of a function what we mean by specifying the domain. of a function and absolute value function,1 1 What is a function. 1 1 1 De nition of a function, A function f from a set of elements X to a set of elements Y is a rule that. assigns to each element x in X exactly one element y in Y. One way to demonstrate the meaning of this de nition is by using arrow diagrams. f X Y is a function Every element g X Y is not a function The ele. in X has associated with it exactly one ment 1 in set X is assigned two elements. element of Y 5 and 6 in set Y, A function can also be described as a set of ordered pairs x y such that for any x value in.
the set there is only one y value This means that there cannot be any repeated x values. with di erent y values, The examples above can be described by the following sets of ordered pairs. F 1 5 3 3 2 3 4 2 is a func G 1 5 4 2 2 3 3 3 1 6 is not. tion a function, The de nition we have given is a general one While in the examples we have used numbers. as elements of X and Y there is no reason why this must be so However in these notes. we will only consider functions where X and Y are subsets of the real numbers. In this setting we often describe a function using the rule y f x and create a graph. of that function by plotting the ordered pairs x f x on the Cartesian Plane This. graphical representation allows us to use a test to decide whether or not we have the. graph of a function The Vertical Line Test, Mathematics Learning Centre University of Sydney 2. 1 1 2 The Vertical Line Test, The Vertical Line Test states that if it is not possible to draw a vertical line through a. graph so that it cuts the graph in more than one point then the graph is a function. This is the graph of a function All possi This is not the graph of a function The. ble vertical lines will cut this graph only vertical line we have drawn cuts the. once graph twice,1 1 3 Domain of a function,For a function f X Y the domain of f is the set X.
This also corresponds to the set of x values when we describe a function as a set of ordered. If only the rule y f x is given then the domain,is taken to be the set of all real x for. which the function is de ned For example y x has domain all real x 0 This is. sometimes referred to as the natural domain of the function. 1 1 4 Range of a function, For a function f X Y the range of f is the set of y values such that y f x for. some x in X, This corresponds to the set of y values when we describe a function as a set of ordered. pairs x y The function y x has range all real y 0,a State the domain and range of y x 4. b Sketch showing signi cant features the graph of y x 4. Mathematics Learning Centre University of Sydney 3. a The domain of y x 4 is all real x 4 We know that square root functions are. only de ned for positive numbers so we require that x 4 0 ie x 4 We also. know that the square root functions are always positive so the range of y x 4 is. all real y 0,4 3 2 1 0 1,The graph of y x 4, a State the equation of the parabola sketched below which has vertex 3 3.
2 0 2 4 6 8,b Find the domain and range of this function. a The equation of the parabola is y 3, b The domain of this parabola is all real x The range is all real y 3. Sketch x2 y 2 16 and explain why it is not the graph of a function. x2 y 2 16 is not a function as it fails the vertical line test For example when x 0. Mathematics Learning Centre University of Sydney 4. The graph of x2 y 2 16,Sketch the graph of f x 3x x2 and nd. a the domain and range,The graph of f x 3x x2, a The domain is all real x The range is all real y where y 2 25. b f q 3q q 2, Mathematics Learning Centre University of Sydney 5.
c f x2 3 x2 x2 3x2 x4,f 2 h f 2 3 2 h 2 h 2 3 2 2 2. 6 3h h2 4h 4 2, Sketch the graph of the function f x x 1 2 1 and show that f p f 2 p. Illustrate this result on your graph by choosing one value of p. The graph of f x x 1 2 1,f 2 p 2 p 1 2 1, Mathematics Learning Centre University of Sydney 6. The sketch illustrates the relationship f p f 2 p for p 1 If p 1 then. 2 p 2 1 3 and f 1 f 3, 1 2 Specifying or restricting the domain of a function. We sometimes give the rule y f x along with the domain of de nition This domain. may not necessarily be the natural domain For example if we have the function. y x2 for 0 x 2, then the domain is given as 0 x 2 The natural domain has been restricted to the.
subinterval 0 x 2, Consequently the range of this function is all real y where 0 y 4 We can best. illustrate this by sketching the graph,The graph of y x2 for 0 x 2. Mathematics Learning Centre University of Sydney 7. 1 3 The absolute value function, Before we de ne the absolute value function we will review the de nition of the absolute. value of a number, The Absolute value of a number x is written x and is de ned as. x x if x 0 or x x if x 0, That is 4 4 since 4 is positive but 2 2 since 2 is negative.
We can also think of x geometrically as the distance of x from 0 on the number line. More generally x a can be thought of as the distance of x from a on the numberline. Note that a x x a,The absolute value function is written as y x. We de ne this function as, From this de nition we can graph the function by taking each part separately The graph. of y x is given below,y x x 0 1 y x x 0,The graph of y x. Mathematics Learning Centre University of Sydney 8. Sketch the graph of y x 2,For y x 2 we have,x 2 when x 2 0 or x 2. x 2 when x 2 0 or x 2,x 2 for x 2,x 2 for x 2,Hence we can draw the graph in two parts.
y x 2 x 2 y x 2 x 2,The graph of y x 2, We could have sketched this graph by rst of all sketching the graph of y x 2 and. then re ecting the negative part in the x axis We will use this fact to sketch graphs of. this type in Chapter 2,1 4 Exercises,1 a State the domain and range of f x 9 x2. b Sketch the graph of y 9 x2,2 Given x x2 5 nd in simplest form h 0. 3 Sketch the following functions stating the domain and range of each. Mathematics Learning Centre University of Sydney 9. 4 a Find the perpendicular distance from 0 0 to the line x y k 0. b If the line x y k 0 cuts the circle x2 y 2 4 in two distinct points nd the. restrictions on k, 5 Sketch the following showing their important features. 6 Explain the meanings of function domain and range Discuss whether or not y 2 x3. is a function, 7 Sketch the following relations showing all intercepts and features State which ones.
are functions giving their domain and range,8 If A x x2 2 1. x 0 prove that A p A p1 for all p 0, 9 Write down the values of x which are not in the domain of the following functions. a f x x2 4x,b g x x2 1,10 If x log x 1,nd in simplest form. 11 a If y x2 2x and x z 2 2 nd y when z 3,b Given L x 2x 1 and M x x2 x nd. Mathematics Learning Centre University of Sydney 10. 12 Using the sketches nd the value s of the constants in the given equations. 13 a De ne a the absolute value of a where a is real. b Sketch the relation x y 1,14 Given that S n n,nd an expression for S n 1.
Hence show that S n S n 1 1, Mathematics Learning Centre University of Sydney 11. 2 More about functions, In this Chapter we will look at the e ects of stretching shifting and re ecting the basic. functions y x2 y x3 y x1 y x y ax x2 y 2 r2 We will introduce the. concepts of even and odd functions increasing and decreasing functions and will solve. equations using graphs,2 1 Modifying functions by shifting. 2 1 1 Vertical shift, We can draw the graph of y f x k from the graph of y f x as the addition of. the constant k produces a vertical shift That is adding a constant to a function moves. the graph up k units if k 0 or down k units if k 0 For example we can sketch the. function y x2 3 from our knowledge of y x2 by shifting the graph of y x2 down. by 3 units That is if f x x2 then f x 3 x2 3,1 1 y x 2 3.
We can also write y f x 3 as y 3 f x so replacing y by y 3 in y f x also. shifts the graph down by 3 units,2 1 2 Horizontal shift. We can draw the graph of y f x a if we know the graph of y f x as placing the. constant a inside the brackets produces a horizontal shift If we replace x by x a inside. the function then the graph will shift to the left by a units if a 0 and to the right by a. units if a 0, Mathematics Learning Centre University of Sydney 12. For example we can sketch the graph of y x 2 from our knowledge of y x1 by shifting. this graph to the right by 2 units That is if f x x1 then f x 2 x 2. 2 1 0 1 2 3 4 x,Note that the function y x 2, is not de ned at x 2 The point 1 1 has been shifted. 2 2 Modifying functions by stretching, We can sketch the graph of a function y bf x b 0 if we know the graph of y f x. as multiplying by the constant b will have the e ect of stretching the graph in the y. direction by a factor of b That is multiplying f x by b will change all of the y values. proportionally, For example we can sketch y 2x2 from our knowledge of y x2 as follows.
1 0 1 1 0 1,The graph of y x2,The graph of y 2x2 Note all the y. values have been multiplied by 2 but the,x values are unchanged. We can sketch the graph of y 12 x2 from our knowledge of y x2 as follows.

## Related Books

###### Barnfeet 5 Year Business Plans - Sustainability Partnerships

5 Year Business Plan ... Earning an equivalent amount of 8 hours 5 days a week work or more doing Farming, Is it ... Business Identification for ...

###### ORGANIC FARMING SCHEME BUSINESS PLAN

ORGANIC FARMING SCHEME BUSINESS PLAN Name: Address: Herd No Organic Licence No Date: Previously in SM6 YeNo s

###### A Proposal for a Rwanda Potato Sector Development Program ...

Rwanda Potato Sector Development Program Agricultural Policy Development Project ... action plan for the potato sector.

###### DIFFERENTIAL GEOMETRY: A First Course ... - University of ...

DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Preliminary Version Summer, 2016 Theodore Shifrin University of Georgia Dedicated to the memory of Shiing ...

###### Dissertation - Active converter based on the VIENNA Rectifie

ACTIVE CONVERTER BASED ON THE VIENNA RECTIFIER TOPOLOGY INTERFACING A THREE-PHASE GENERATOR TO A DC-BUS by Jacobus Hendrik Visser Submitted in partial fulfillment of the requirements for the degree

###### Public Private Partnerships for Impacting Poor Community

Public Private Partnerships for Impacting Poor Community Roundtable Proceedings This document captures the highlights of thoughts and solutions on promoting

###### Business for Development - Sida

Business for Development ... works in a systematic way in partnerships with the business sector. ... We know that many companies do a great deal of good for poor people

###### Hinweis Bei dieser Datei handelt es sich um ein Protokoll ...

Hinweis Bei dieser Datei handelt es sich um ein Protokoll, das einen Vortrag im Rahmen des Chemielehramtsstudiums an der Uni Marburg referiert. Zur besseren

###### Repair and Rehabilitation of Zone Five Tendon Injuries of ...

Repair and Rehabilitation of Zone Five Flexor Tendon Injuries of the Wrist Galal Hegazy* 1, Ahmed Akar , Emad Zayed1, Mohamed Ellabad2 and Ahmed Mosalam3 1Orthopedic Department, Al-Azhar University, Egypt 2Nurosurgery Department, Al-Azhar University, Egypt 3physical medicine Rheumatology and Rehabilitation Department