3.9

© 2009 Chung Tai Educational Press. All rights reserved.

Each of the following figures represents a relation between x and y. Determine whether y is a function of x. If yes, write down its domain and co-domain. (1 − 2)

1. (a) 123

4710

(b) 6

12

111315

2. (a) 1257

36912

(b) 57911

1248

For each of the following tables, determine whether y is a function of x. Explain briefly. (3 − 4)

3. (a) x 0 1 2 3

y 4 3 2 1 (b) x 1 3 5 7

y 2 2 4 4

4. (a) x 1 2 2 2 3

y 3 4 5 6 8 (b) x 1 4 7 8 9

y 3 3 3 3 4

It is given that a, b and c are three distinct real numbers. For each of the following tables, determine whether y is a function of x. Explain briefly.

5. (a) x a b c

y c b a (b) x a b a

y b c c

3.10

© 2009 Chung Tai Educational Press. All rights reserved.

Draw the graphs of the following functions for the given ranges. (6 − 7)

6. (a) 40 ;2 ≤≤−= xy (b) 22 ;4 ≤≤−+= xxy

7. (a) 51 ;22 ≤≤+−= xxy (b) 31 ;43 ≤≤−−= xxy

For each of the following graphs, determine whether y is a function of x for 44 ≤≤− x . Explain briefly. (8 − 11)

8. (a)

x

y

O 4− 4

(b)

x

y

O 4− 4

9. (a)

x

y

O 4− 4

(b)

x

y

O 4− 4

10. (a)

x

y

O 4− 4

(b)

x

y

O 4− 4

11. (a)

x

y

O 4− 4

(b)

3.11

© 2009 Chung Tai Educational Press. All rights reserved.

x

y

O 4− 4

In each of the following, y is a function of x. Find the largest domain of real numbers for each function. (12 − 15)

12. (a) xy 2−= (b) 22 += xy

13. (a) 5+= xy (b) 2−= xy

14. (a) 83 += xy (b) 69 −−= xy

15. (a) 3

1+

=x

y (b) 24

1 +−

=x

y

Write down the largest domain of real numbers for each of the following functions. (16 − 17)

16. (a) 213

2 +

−=xxy (b)

1234

2 +

+=

xxy

17. (a) x

xy6334

−

+= (b)

25)3)(2(

−−

−+=

xxxxy

For each of the following algebraic expressions, determine whether y is a function of x for 0>x . Explain briefly. (18 − 21)

18. (a) 2+= xy (b) 34 −= xy

19. (a) 322 += xy (b) )25(2 += xxy

3.12

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20. (a) 14

1 −−

=x

y (b) 2

1+

=x

y

21. (a) 526+

+=

xxy (b)

2

152 +

−=

x

xy

22. Given that 2)4()( −= xxg , find the values of the following. (a) )0(g (b) )4(g (c) )4(−g

23. Given that 31)(−

=x

xh , where 3≠x , find the values of the following.

(a) )2(h (b) )0(h (c) )3(−h

24. Given that 32

3)(

+−

=x

xf , where 23

−≠x , find the values of the following.

(a) )2(2 f− (b) )21(3 f× (c) )3(21 −×+ f

25. Given that 1)( 2 += xxp , find the values of the following.

(a) )1()2( pp − (b) )2()3( pp × (c) 2)]2([ −p

26. Given that 3)( −= xxf and x

xg 1)( = , where 0≠x , find the values of the following.

(a) )41()4( gf + (b) )

21()5( gf × (c)

)2()2(

gf

27. Let 2)( 2 += xxh . Determine whether each of the following expressions holds.

(a) )2()1()21( hhh +=+ (b) )2()1()21( hhh ×=× (c) 22 )]2([)2( hh =

3.13

© 2009 Chung Tai Educational Press. All rights reserved.

28. Given that 43)( 2 += xxf , find (a) )2( nf . (b) )2( +nf . (c) )13( −nf .

29. Given that 694)( 2 −+= xxxg , find (a) )3( +ng . (b) )32( −ng . (c) )33( −− ng .

30. Given that 12)( += kxxf and 10)10( =f , where k is a constant, find the value of k.

31. It is given that 62)( 2 −+= xxxf . If kkf =)( , find the values of k.

32. It is given that kxxxg ++= 3)( 2 and 3)4( −=−g , where k is a constant. Find the value of )5(−g .

33. It is given that kxkxkxf 3)4()1()( 2 −−+−= , where k is a constant. If )23()1( −=− ff , find the value

of )3(f .

34. It is given that 10)8(5)( 2 −++= xkxxg , where k is a constant and 0>k . If kkg =)( , find the value of )6(−g .

35. It is given that )74()65(6)( 2 −+−+= kxkkxxf , where k is a constant. If 20)4()2( =− ff , find the value of )3(f .

36. Let qpxxf +=)( , where p and q are constants. It is given that 1)10( −=f and 2)14( −=f .

(a) Find the values of p and q.

(b) Solve the equation 25)84( =+xf .

37. Let bxxaxf +−−= )2)(12()( , where a and b are constants. It is given that 5)1( −=f and 13)3( =f .

3.14

© 2009 Chung Tai Educational Press. All rights reserved.

(a) Find the values of a and b.

(b) Solve the equation 05)1( =+−xf .

38. It is given that 45)( −= xxf and 62)( +−= xxg . If 42)()3( =+ xgxf , find the values of x.

39. It is given that 16)( 2 +−= xxxf and 542)( 2 ++= xxxg . If 7)2()32( =+++ xgxf , find the value of x.

40. In the figure, ABCD is a square cardboard with sides of 16 cm each. A square with sides of x cm each is cut from each corner of the cardboard. Then the cardboard is folded up to form a box without a lid.

16 cmBA

CD

x cm

x cm (a) Express the volume of the box as a function )(xf (in

3cm ).

(b) Find the volume of the box when 3=x .

41. In a factory, the total cost of producing n boxes of models is )(nC (in $), where nnC 30000 25)( += .

(a) Find the total cost of producing 1 500 boxes of models.

(b) It is given that the selling price of each box of model is $120. Express the profit of selling n boxes of models as a function )(nP (in $).

(c) If 2 000 boxes of models are sold, find the profit of the factory.

42. The fee of a trip with n students participated is )(nE (in $), where nnE 30600)( += . (a) If 30 students participate in the trip and the fee is evenly shared, find the amount paid by each

student. (b) Among the 30 students, 5 of them cannot afford the fee and the remaining 25 students decide to

share the fee of those 5 students evenly. Find the amount paid by each of the 25 students.

43. It is given that xxxxf −+= 35 2)( . (a) Prove that )()( xfxf −=− .

3.15

© 2009 Chung Tai Educational Press. All rights reserved.

(b) Hence find the value of )199()199( −+ ff .

44. It is given that )52)(10()( 22 −−= xxxf . (a) Prove that )()( xfxf −= . (b) Hence find the value of )3(99)3(100 ff −− .

45. Write down a function )(xf such that kxfkxf +=+ )()( , where k is a non-zero constant.