NJC 03 2016 - 2017 H2 Maths Functions Lecture Questions (Stu) (Final)

National Junior College Mathematics Department 2016 Functions Page 1 of 12 National Junior College 2016 – 2017 H2 Mathematics Functions Lecture Questions Part 1. Key Questions to answer:  What are the set builder and interval notations, and how do you use them? Prerequisite knowledge: Number Line, Set Notation and Language Lecture Readings: Section 1 Question 1.1 What do the following symbols represent? , , , , ,    Answer: : set of integers, : set of positive int egers, : set of negative integers, : set of real numbers, : set of positive real numbers, : set of negative real numb ers   Question 1.2 What are set operators and how are they used as an extension to the above sets? Question 1.3 How do we use the set builder notation or the interval notation to represent a set of values? Range of Values Representation as a Set Set Builder Notation Interval Notation 23 x    | 2 3xx    ( 2, 3) 23 x    | 2 3xx    [ 2, 3) 23 x   2x  |2xx  ( , 2]  3x Question 1.4 What is the difference between the use of round and square brackets in the interval notation? Part 2. Key Questions to answer:  What are relations and functions?  How do you determine graphically whether a relation is a function?  How do you define and represent a relation/function? Prerequisite knowledge: Curve Sketching Lecture Readings: Section 2

National Junior College Mathematics Department 2016 Functions Page 2 of 12 Question 2.1 (Numerical representation of relations or functions) Which of the following relations is a function? Function Not a Function Function Question 2.2 (Diagrammatical representation of relations or functions) Which of the following relations is a function? Learning Point(s): (a) A relation is an association between 2 sets of data (in our case, they are usually numbers). (b) A function is a relation that has exactly one output for every possible input [colloquially we think of it as many-to-one (function) versus one-to-many (not a function)]. Question 2.3 (Algebraic representation of relations or functions) Which of the following relations is a function? h : , {0, 4},x x x g : , { 4, 0},x x x  f : , {0, 4},x x x  p( ) .xx  The relations g and h are not functions. Why?  The relation f is a function. Note that  f xx is known as the rule of function f. {0, 4} is the domain (“inputs”) of f, denoted by fD . {0, 2} is the range (“outputs”) of f, denoted by fR . A function is defined by both its rule and domain.  If the domain of a function is not given explicitly, the convention is that the domain is the set of all numbers for which the rule makes sense and defines a real number. In the case of function p, the domain of p is [0, ). The corresponding range of p is [0, )pR  .  Functions p and f are not the same even though they have the same rule , as they have different domains. x yx 2 2 2 2 1 1 1 1 22 4yx x y 0 2 1 3 2 0 x 1yx 0 1 1 2 2 3 3 4 Y X 1 2 3 4 3 5 7 9 Function Y X Function 1 2 3 4 3 5 7 9 Y X Not a Function 1 2 3 4 3 5 7 9

National Junior College Mathematics Department 2016 Functions Page 3 of 12 Question 2.4 (Graphical representation of relations or functions) Which of the following relations is a function? In the case where a relation is a function, state the range of the function. f : , [0, )x x x  g : , [0, )x x x   h : sin( ), [0,2 ]x x x  Relation f is a function. f [0, ).R  Relation g is not a function. Relation h is a function. h [ 1, 1].R  Learning Point(s): (a) When a function is represented graphically as a curve, the x-values on the curve belong to the domain of f, and the y-values belong to the range of f. (b) Relevant graphs must be shown when using the vertical line test. (c) To find the range of a function, you need to know the key features of its corresponding graph. In particular, checking the y-coordinates of the end-points of a curve is not sufficient. _________________

National Junior College Mathematics Department 2016 Functions Page 4 of 12 Question 2.5 (i) Show that 22 4xy , where ,xy  , is not a function. (ii) Define the function f such that all the following conditions are satisfied: (1) any point with coordinates  , f( )xx lies on the curve 22 4,xy (2) the domain of f is maximal, and (3) the range of f is a subset of , the set of all negative real numbers. (iii) Prove that your choice of f is indeed a function, and state the range of f. Solution: (i) From the graph of 22 4xy , the line x = 1 cuts the graph twice. Hence 22 4xy is not a function. (ii) 2f : 4 , ( 2, 2).x x x    (iii) From the graph of 2f( ) 4y x x   , every vertical line x = a, where ( 2, 2)a , cuts the graph once. Hence, f is a function. f [ 2,0).R  Learning Point(s): (a) When defining a function, we need the rule and domain stated clearly. (b) You may express it as ff : ..., .x x D  (c) The set of negative real numbers does not include 0. 2 –2 –2 2 2 –2 –2 1x

National Junior College Mathematics Department 2016 Functions Page 5 of 12 Part 3. Key Questions to answer:  What is a one-one function?  How do you test whether a function is one-one? Prerequisite knowledge: Curve Sketching, Piecewise and Quadratic expressions Lecture Readings: Section 3 Question 3.1 Based on the table below, what makes a function a one-one function? One-one function Not a one-one function Set Diagram (Distinct points) Set Diagram (Distinct points) Graphical Representation (Continuous interval of points) Learning Point(s): (a) A function f is said to be one-one (1-1) or injective if no two distinct elements in its domain have the same image under f. Question 3.2 Graphically, how do we test if a function is one-one? Solution: The horizontal line test. 1 2 3 4 3 5 7 9 1 2 3 4 3 5 7 9

National Junior College Mathematics Department 2016 Functions Page 6 of 12 Learning Point(s): (a) To prove that a function is one-one, we need to show that every horizontal line f,y b b R cuts the graph of ff( ),y x x D exactly once, or every horizontal line ,y b b cuts the graph of ff( ),y x x D at most once. In this case, it is necessary to provide a relevant graph to support your statement. (b) To prove that a function is not one-one, we need to find one horizontal line such that it cuts the graph of ff( ),y x x D at least two times. Question 3.3 Justify if 2f : 4 1, [1, )x x x x     is a one-one function. If it is not one-one, find a restriction of f such that it is one-one. Solution: f is not one-one, because the line 2y cuts the curve y = f(x) at two points. Alternatively to prove or disprove that a function is one-one, you may use the algebraic definition for one-one functions (self-reading). Observe that if we restrict the domain of f to [1, 2] or [2, ) , then the function with the restricted domain is one-one. Question 3.4 Determine if the following are one -one functions, giving your reasons. If it is not one -one, find a restriction of the function with a maximal domain such that it is one -one and has the same range as the original function. (i) 1 0,f( ) 0, xxx ax x   (ii) g : , ,x x b x (iii) 2h : 2 1, [ , ),x x cx x c    where , anda b c are positive constants. Solution: (i) Not one-one. Use the line 1y Restrict domain to  ,0 . (ii) Not one-one. Use the line