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.

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