TMJC H2 Chapter 3 Functions_Discussion Questions_Solutions_2024

Chapter 3 Functions TMJC 2024 Page 1 of 20 H2 Mathematics (9758) Chapter 3 Functions Discussion Questions (Suggested Solutions) Level 1 1 Sketch the graphs of each of the following functions. State its domain and give its corresponding range. (a) 23f : , , 1 1 xx x x x − − (b) ( ) 2 g : 2 4, , 2x x x x− − +   (c) ( )h : ln 1 , ,1 3x x x x −    1 Solution (a) ( )fD 1,= ( )fR , 2= − (b) gD ( ,2]= − gR ( ,4]= − y x O y = f (x) y x y = g(x) 23 1 xy x −= − Vertical asymptote: 1x= Horizontal asymptote: 2y= Find by referring to the graph of f to determine the range of possible y values. (i.e. minimum and maximum y values) Take the domain of f into consideration when sketching the graph of f ( )2, 4 Use a closed circle at the point since is included in the domain of g.

Chapter 3 Functions TMJC 2024 Page 2 of 20 (c) ( )hD 1,3= ( )hR ,ln 2= − y x O y = h(x) Use an open circle at the point since is excluded from the domain of h. ( )3,ln 2 Ensure the graph approaches the vertical asymptote

Chapter 3 Functions TMJC 2024 Page 3 of 20 2 The function f is defined by ( ) 2 f : 2 4,xx − − + , 0xx . (i) Sketch the graph of f. (ii) State the domain and range of f. (iii) Explain why 1f− does not exist. 2 Solution (i) (ii) ( )fD 0,= ( fR , 4= − (iii) Method 1: Horizontal Line Test Since the horizontal line 3y= cuts the graph of ( )fyx= more than once, f is not a one-one function. Hence 1f− does not exist. Method 2: Algebraic Method Since ( ) ( )f 1 f 3 3== , f is not a one-one function. Hence 1f− does not exist. y x 3y= Check both the rule and domain when sketching the graph. Open circle since 0x= is not included. Show that f is not a one-one function by giving a specific example of a horizontal line that cuts the graph the graph more than once. Remark: Any horizontal line yk= with 04 k will be accepted. To show that inverse does not exist, show that f is not a one-one function Show that f is not a one-one function by giving a specific example of two inputs that result in the same output.

Chapter 3 Functions TMJC 2024 Page 4 of 20 3 The function f is defined as 23f : , , 1 1 xx x x x − − . (i) Explain why 1f− exists. (ii) Find 1f− in a similar form. 3 Solution (i) Since any line ya= , where a , cuts the graph of f at most once, f is one-one. Hence 1f− exists. (ii) Let 23 1 xy x −= − ( ) ( ) 1 2 3 23 3 2 y x x x y y yx y − = − − = − −= − ( ) ( ) 1 1 3f 2 3f 2 yxy y xx x − − −== − −= − ( )1 ffD R , 2− = = − 1 3f : , 2,2 xxx x x− − −  y x O y = f (x) ya= To explain that 1f− exists  show f is one-one (horizontal line test) Note: you must draw the graph if it is not given by the question. To write the expressi