chapter 1 functions paper 1 with answer

7/23/2019 Chapter 1 Functions Paper 1 With Answer

http://slidepdf.com/reader/full/chapter-1-functions-paper-1-with-answer 1/39

Additional Mathematics Chapter 1 Functions

1. H = { 3, 5, 6 }

K = { 5, 7, , !, 1" }

#ased on the a$o%e in&ormation, the relation $et'een H and K is de&ined $( the set o& ordered pairs

{ )3,5*, )3,7* , )5,*, )5,!* }+

tate

)a* the ima-e o& 3,)$* the o$.ect o& 5+

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

2. ia-ram sho's the relation $et'een set 2 and set +

tate

)a* the ran-e o& the relation,

)$* the t(pe o& the relation

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



3. ia-ram sho's the relation $et'een set 4 and set +

tate

)a* the t(pe o& the relation,

)$* the o$.ect &or &+

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

4. i%en that 3 +→ x f:x and 15" +−→ x x g:x , &ind

)a* ),( f - 71

)$* in similar &orm the &unction + fg

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

5.i%en the &unctions k x f:x +→ 6 and

1 73

1 f : x px ,

− → + 'here k and p are constants,

&ind the %alue o& k and o& p+

Ans'er / k = 0000000000000000000000

p = 0000000000000000000000



6.i%en the &unction ,

x f(x)

= 8≠ x and the composite &unction x, fg(x) = &ind

)a* g(x),

)$* the %alue o& x 'hen gf(x) = 7

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

7. i%en 5" x f(x) += and x g(x) −= 3 , &ind 1− gf +

Ans'er / 0000000000000000000000000


x f:x

3"

"8

−→ , x

"

3≠ and k, f(k) = &ind the %alue o& k +

Ans'er / 0000000000000000000000000


x

x f:x

"5

−→ , x

5

"≠ e%aluate +31 )( f -

Ans'er / 0000000000000000000000000



10. i%en that , x f(x) 5" −= &ind *)11 x f f -- to its simplest &orm+

Ans'er / 0000000000000000000000000

11.i%en that ,

k

x - f : x

"

3"→ ,89 ≠ and 8

351 , mm

x:x f - ≠

+→ , 'here k and m are

constants+ Find the %alues o& k and o& m+

Ans'er / k = 0000000000000000000000

m = 0000000000000000000000

12.i%en that ,

x- ) x - k, g(x f(x)

5

3 == and

5

7−=

mx fg(x) 'here k and m are constants+

Find the %alues o& k and o& m+

Ans'er / k = 0000000000000000000000

m = 0000000000000000000000

13. A &unction f is de&ined $( q. px f : x +→ :he ima-es o& 1 and 5 are ;" and 18

respecti%el(+ Calculate the %alue o& p and o& q+

Ans'er / p = 0000000000000000000000

q = 0000000000000000000000



14. A &unction f is de&ined $( . x f : x 15 −→ Find

)a* the possi$le %alue o& x 'hose ima-e is ;11,

)$* the ran-e correspondin- to a domain o& . x 31 ≤≤

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

15. A &unction f is de&ined $( +"" x x f : x −→ Find

)a* the possi$le %alues o& x 'hose ima-e is 3,

)$* the ran-e correspondin- to a domain o& . x "1 ≤≤−

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

16. A &unction & is de&ined $( . x x f : x 55" −+→ Find

)a* the ima-e o& 3 under f ,

)$* the possi$le %alues o& < 'hich are unchan-ed $( mappin-+

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



17.A &unction f is de&ined $( ,

x x f : x

53 +→ . x 8≠ Calculate

)a* the ima-e o& 5 under &,

)$* the possi$le %alues o& < 'hose ima-e is +

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

18.i%en that 8" ≠+→ ,x

x

p f : x and that "

"

11 −= )( f -

,&ind

)a* the %alue o& p,

)$* the elements 'hich are unchan-ed under f +

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

19.i%en the &unctions x f:x −→ " and , ,x

x g : x 8

3≠→ &ind in similar &orm,

)a* ,1−

f

)$* fg +

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



20.i%en the &unctions x f : x 3 −→ and , , x

x g : x 8

5" ≠−→ &ind in similar &orm,

)a* ,1- f

)$* fg.

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

21.A &unction f is de&ined $( . ,x

x

x f : x 1

1≠

−→ <press in similar &orm, the &unction +" f

Ans'er / 0000000000000000000000000

22. Functions & and - are de&ined $( x f : x 3 −→ and ,k hx g: x +→ 'here 8+h >Find the %alues o& h and k 'hich ( ) ( ) x f x g =" &or all %alues o& <+

Ans'er / 0000000000000000000000000


x

x - f : x

3

1"

+→ +3−≠ x Find " f in similar &orm+

Ans'er / 0000000000000000000000000



24. Functions f and g are de&ined $( 1" +→ x f : x and .- x g : x 1"→)a* <press in similar &orm the &unction gf ,

)$* Find the %alues o& x 'hich gf = g.

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

25.A &unction , x ,

x-

x f : x 1

1

"≠→

)a* <press in similar &orm the &unction ," f

)$* tate the %alue o& x &or 'hich the &unction " f is not de&ined+

Ans'er / 0000000000000000000000000

26.

Functions f and g are de&ined $( 1" +→ x f : x and 11 ≠−→ x ,

x x g : x +

)a* Find the %alue o& ( ) ,- f 51

)$* <press in similar &orm the &unction gf.

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



27.A &unction is de&ined $( ,

x f : x

1"

+→

"

1−≠ x and ,

p x

kx : x f

++

→"

" ,

p x

"−≠ 'here

k and p are constants+ Find the %alues o& k and o& p+

Ans'er / k = 0000000000000000000000

p = 0000000000000000000000

28. :he &unction f is de&ined $( 1−→ x f : x and &unction fg is de&ined $(

. x x fg : x 5"" ++→ <press the &unction o& g +

Ans'er / 0000000000000000000000000

29. :he &unction f is de&ined $( 1" +→ x f : x and &unction &- is de&ined $(

. x , x

fg : x 13

−≠

+→ <press the &unction o& g +

Ans'er / 0000000000000000000000000

30. :he &unction f is de&ined $( 1+→ x f : x and &unction gf is de&ined $(

. x x gf :x 5"" ++→ <press the &unction o& g +

Ans'er / 0000000000000000000000000



31. :he &unction f is de&ined $( "+→ x f :x and &unction gf is de&ined $(

. x x gf :x 5" ++→ <press the &unction o& g +

Ans'er / 0000000000000000000000000

32.A &unction f is de&ined $( , xk,

x f : x 8

"≠+→ 'here 9 is a constant+ i%en that

( ) ( ),13

""" f f = calculate the possi$le %alues o& k +

Ans'er / 0000000000000000000000000

33.A &unction f is de&ined $( . x ,

x-

x f : x "

"≠→ Find the %alue o&

)a* &)6*,

)$* ( )+"& >1

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



34.A &unction f is de&ined $( . x

x-

x f:x " ,

"

1≠

+→

)a* Find in similar &orm ,1− f

)$* tate the %alue o& x 'hich 1− f is not de&ined+

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000


x

x f:x "

"

!≠

−−

→ Find

)a* the %alue o& ( )1 1 f ,− −)$* the %alue o& x 'hich ( ) x. x f - =1

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000


x

x f:x 1

1

""≠

−+

→

)a* Find the %alue o& ( ) , f 31−

)$* i%en that ( ) ,1 mpm f =− e<press p in terms o& m+

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



37.A &unction f is de&ined $( +8

"6 >−→ x ,

x x f:x Find

)a* the %alue o& ( ) , f 1−

)$* the %alue o& 9 i& ( ) "1 −=− k f

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

38. A &unction f is de&ined $( . x f:x 1" −→)a* 9etch the -raph o& f &or the domain +11 ≤≤− x

)$* Find the ran-e o& f +

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

39. A &unction & is de&ined $( k.h x f:x +−→ i%en that ( ) 33 = f and ( ) ,31 =− f &ind the

%alue o& h and o& k +

Ans'er / h = 0000000000000000000000

k = 0000000000000000000000



40. :he &unction k hx x f:x ++→ 3 is such that the e?uation ( ) x x f = has solutions o& x = "

and x = 3+ Find the %alue o& h and o& k +

Ans'er / h = 0000000000000000000000

k = 0000000000000000000000


x

x f:x

18≠

−+

→

)a* Find the %alue o& ( ) , f 51−

)$* A positi%e num$er 9 such that ( ) k.k f =

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

42. :he &unction f is de&ined $( 3 +→ x f:x and &unction g is de&ined $( . x g:x 6" +→Find the %alue o&

)a* ( ) , fg 3−

)$* ( ). gf 11−

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



43.A &unction f is de&ined $( "

"

3−≠

+−

→ x , x

x f:x + i%en that ( ) ,

"

11 =−k f &ind the %alue o&

9+

Ans'er / 0000000000000000000000000


1

"3≠

−+

→ x , x

x f:x Find the %alue o&

)a* ( ) , f f "1−

)$* ( ). f 1−

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

45.

i%en that8>+→ pq, px f:x

and . x:x f 1816

"

−→ Find)a* the %alue o& p and o& q,

)$* the %alue o& ( ). f 11−

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



46.i%en that x ,

x

k x f:x "

"≠

−+

→ and that f )7* = "+ Find the %alue o&

)a* 9,

)$* ( ). f 1 −−

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

47. :he &unction f is de&ined +3 x f:x −→ Find an e<pression, in terms o& < &or

)a* ( ) , x f 1−

)$* ( ). x f "

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

48.Functions f and g are de&ined $( x ,

x f:x 8

3

1≠→ and +3−→ x g:x Find an e<pression,

in terms o& x &or

)a* ( ) , x f "

)$* fg(x)+

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



49. i%en that ( )"1−→ x f:x &or domain ,58 ≤≤ x &ind the ran-e o& f +

Ans'er / 0000000000000000000000000

50. Functions f and g are de&ined $( " f : x x h,→ + 3 g : x kx→ − and 518 −→ x fg:x

'here h and k are constants+ Find the %alue o& h and o& k.

Ans'er / h = 0000000000000000000000

k = 0000000000000000000000

51. i%en A= { }3,",1, and set # is the set o& all inte-ers, dra' the arro' dia-ram sho'in-

the &unction 3" x - f:x → 'here A+< ∈

Ans'er / 0000000000000000000000000

52. @& "+→ x f:x &rom set A to set # and 1−→ x g:x &rom set # to set C, illustrate 'ith an

arro' dia-ram, the composite &unction -& &rom set A = { }3,",1, +

Ans'er / 0000000000000000000000000



53.Functions f and g are de&ined $( 3" +→ x f:x and +

13≠

−+

→ , x x

x g:x Find the %alue

o& ( )1 5 + gf −

Ans'er / 0000000000000000000000000

54.A &unction f is de&ined $( ,

!

!

k x ,

kx f:x

−≠

+→ and +"*1) = f Find

)a* the %alue o& 9,

)$* the ima-e o& ;1+

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000


1

1≠

−

+→ x ,

x

x f:x <press in their simplest &orms

)a* ( ),<& "

)$* ( )+<&

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



56.A &unction f is de&ined $( ,1

1−≠

+→ x ,

x

x f:x and ,

1"

"

+→

x

x:x f

"

1−≠ x + Find

)a* the %alue o& ( ) , f 1" −

)$* a similar e<pression &or . f 3

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

57. i%en the 13 −→ x f:x and +7+→ x g:x Find

)a* the ima-e o& under f ,

)$* the %alue o& ( ). g f 31−

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

58.i%en the &unctions 3

"+→

x f:x and ,

x g:x "

3−→ &ind

)a* the %alue o& *,")− gf

)$* the %alue o& x &or 'hich +1*) −= x fg

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000



59. @& p x f:x +→ 3 and xq g:x 3" −→ such that fg = gf , &ind the relationship $et'een p

and q+

Ans'er / 0000000000000000000000000

60.A &unction 1− f is de&ined $( ,

x

x:x f

3

5"1

−−

→− ,3≠ x &ind

)a* the %alue o& < &or 'hich ( ) , x f 81 =−

)$* the &unction, f +

Ans'er / )a*00000000000000000000000

)$*00000000000000000000000

Ans'er /

1))a* 5 and 7

)$* 3+

2)

)a* {',(}

)$* man( to one relation+

3)

)a* man( to man( realation

)$* a+

4)

)a* et k )( f =− 71

so 7= f(k)

739 =+ 9 = 1

)$* fg(x) = 15" +− x x f

= ( ) 315<< " ++−

= 7"8<< " +−

5)



et ( ) y x f =−1

o x f(y) = xk y =+6

6

k x y

−=

( )6

1 k x x f

−=−

1

73

66

1+=− px

k x

1

7

66

13 =

−=

k , p

3

7

1

1−== k p

6)

)a* x fg (x)

= [ ] x g(x) f =

x g(x)

=

x

g(x)"

=

)$* 7= gf(x)

7

=

x

g

7

"=

x

x

56" =

"= x

7)

1et - f (x) y=o f(y) x=" y B 5 = x

"

5−=

x y

"

51 −=

x(x) f

-

1 5

"

5 3

"11

"

- x gf (x) g

x

x

− = −

= −

−=



8)

k f(k) =

k k

=− 3"

"8

8"83"" =−− k k

( )( ) 85" =−+ k k

85" =+k or k ; = 8

"

5−=k or 9 =

9)1et 3- f ( ) k =

o 3 f(k) =

3"5

=−k

k

15 6

16

11

k k

k

= −

=

10)1et

" 5

5

"

51

"

51 1 1

"

- f (x) y

f(y) x

y x

x y

x f (x)

x f f (x) f

==

− =

+=

+− =

+ − − −= 5

5"

"

5 18

15

x

x

x

++

=

+ +=

+=

11)



1

" 3

"

"( 3 = "9<

" 3

"

- Let f (x) y

f(y) x

y x

k

kx

y

==−

=

−+

=1 " 3

"

" 3 5 3

"

" 9 = 5 or m = "

5

"

- kx f (x)

kx x

m

k

+=

+ +=

=

12)

−

=5

x f fg(x)

1

751"3

5

7

5

51"3

5

51"3

5

3

- k

-k - - orm

mxk x

k x

k x-

===

−=

−−

−−=

−

=

13)

( )

>5?

>"?3

3 p>1"p>

)"*)1*

)"*18?5p&)5*

1"? p&)1*

? p<&)<*

==+

= =

−→=+=

→−=+=+=

14)

)a*

11

5 1 11

"

f(x)

x

x

= −− = −

= −

)$*



1

11353

1151

≤≤=−=

=−=

f(x)

)( ) f(

)( ) f(

15)

)a*

( )

"

"

3

" 3

" 3 8

3 1 8

3 8 1 8

3 1

f(x)

x x

x x

x (x )

x or x

x x

=− =

− − =

− + =

− = + == = −

)$*

31

8""""

11"11

8831"11

"

"

"

≤≤−=−=

−=−=

= =−−−=−

f(x)

)( )( ) f(

)( )( ) f(

) f( )( )( ) f(

16)

)a*"3 3 5 3 5

1!

f( ) ( ) ( )= + −=

)$*

15

815

85

55

"

"

or x

) )(x(x

x x

x x x

x f(x)

−==−+

=−+

=−+

=

17))a*

( )

55 3 5

5

16

f( ) ( )= +

=

)$*



( ) ( )

13

5

8153

853

5

3

&

"

ro x

x x

x x

x x

(x)

=

=−−=+−

=+

=

18)

)a*

3

1

"

1

""

"

1"

""

11

==−

=−+

=−

−=−

p

p

p

) f(

)( f

)$*

( ) ( )13

813

83"

3"

"

−==+−

=−−

=+

=

or x

x x

x x

x x

x f(x)

19)

)a*1

1

Aet

"

"

"

f (x) y

f(y) x

y x

y x

f (x) x

−

−

==

− == −

= −

)$*

3

3 "

" 3

fg(x) f x

x

x

x

=

= −

−=

20)

)a*



1

1

et

3

3

3

f (x) y

f(y) x

y x

x y

x

f (x)

−

−

==

− =+

=

+=

)$*

5"

5 " 3

"8 3

5 "8

fg(x) f x

x

x

x

x

= − = − − ÷

= − −

−=

21)

"

1

1

11

1 1 "

1

1 "

x f (x) f

x

x

x x

x x

x x

x

x

x

= −

−=

− −

−=−−

=−

22)

1

3""

3

3

3

"

"

"

−=−=+=−=+=

−=++

−=++=

k

k k ,h

k hk ,h

xk hk xh

xk k)h(hx f(x)(x) g

23)



" " 1

3

" 1" 1

3

" 13

3

" 3

" 1 3 !

3 5

5 5

x f (x) f

x

x

x

x

x

x (x )

x x

x , x

x

− = + − − ÷+ =

−+

+

− − +=− + +− −

= ≠+

24)

)a*

[ ]"

"

" 1

)" 1* 1

gf(x) g x

x

x x

= +

= + −

= +

)$*

13

1

8113

813

1

"

""

−−=

=++=++

−=+

or x

) )(x x(

x x

x x x

25)

)a*

" "

1

""

1

"1

1

" 1

1

x f (x) f

x

x

x

x

x

x x x

x

x

= ÷− ÷− =

−−

= − +

=+

)$*

x B 1 = 8

x = >1

26)

)a*



>1Aet & )>5*=9

o 5

" 1 5

3

f(k)

k

k

= −+ = −

= −

)$*

[ ]" 1

" 1

" 1 1

" 1

"

gf(x) g x

x

( x )

x

x

= ++

=+ −

+=

27)

( )

"

" 1

" 1

" 1

" 1

" 1

" !

" ! "

!

f (x) f x

( ) x

( x )

( x )

x

x

x kx

x x p

k , p

= ÷+

=+

−+

=+ +

+=

++ +

=

+ += =

28)

[ ]

6"

5"1

5"

5"

"

"

"

"

++=

++=−

++=

++=

x x g(x)

x x g(x)

x x g(x) f

x x fg(x)

29)



[ ]

3

3

" 1

3

" 13

3

3

1

3

1

" 3

fg(x) x

f g(x) x

g(x) x

g(x) x

x

x

x

x

x g(x)

(x )

=+

=+

+ =+

= −+− −

=+

−=

+−

=+

30)

[ ]

"

"

"

"

"

"

" 5

1 " 5

1

1

) 1* ") 1* 5

" 1 " " 5

gf(x) x x

g x x x

Let y x

So, x y

g(y) y y

y y y

y

g(x) x

= + +

+ = + +

= += −

= − + − +

= − + + − +

= +

= +

31)

[ ]

"

"

"

"

"

"

) * 5

" 5

"

"

) * ) "* ) "* 5

5

1

) * 1

gf x x x

g x x x

Let y x

So, x y

g y y y

y y y

y

g x x

= + +

+ = + +

= += −

= − + − +

= − + + − += +

∴ = +

32)



( )

"1

8*"*)1)

8"3

""336

3

"

1

"

3

"

1

"

1

"

3

"

"

"

13

"*")

"

""

"

"

or k

k k

k k

k k k k k

k

k

k k

k k

k

k k f

f f

==−−

=+−

+++=++

+=+

++

+=+

+

+=

+

=

33)

)a*6

)6*6 "

3

"

f =−

=

)$*1

1

Aet "

"

"

""

"

f ( ) k

f(k)

k

k k k

k

f ( )

−

−

==

=

−= −=

=

34)

)a*1et ) *

) *

1

"

1 "

f x y

So, f y x

y x

y

y xy x

− ==

+ =−+ = −

1

)1 * " 1

" 1

1

" 1) *

1

" 1

11

-

y x x

x y

x

x f x

x

x

, x x

− = − −− −

=−

− −∴ =

−+

= ≠−)$*

1 > x = 8

x = 1



35)

)a*1et ) 1* f - k − =

o ) * 1 f k = −

1

!1

"! ! "

18 11

11

18

11) 1*

18

k

k k k

k

k

f −

−= −

−− = − +

=

=

− =

)$*

6

86

8361"

36

36

"

!

"

"

"

1

1

==−

=+−

−=−

=−

−

=−

−

=

=

=

−

−

x

)(x

x x

x x x

x

x

x

x x

x

x ) x

f(

x(x) f

x(x) f

36)

)a*1

1

Aet 3

o 3

" "3

1

" " 3 3

5

3 5

f ( ) k

f(k)

k

k

k k

k

f ( )

−

−

==+

=−+ = −

=

∴ =)$*



1

"

"

"

"

" "

1

" "

" ""

"

"

"

f (m) mp

f(mp) m

mpm

mp

mp m p m

p( m m ) mm

pm m

m

m m

− ==

+=

−

+ = −

− = − −− −=

−+

=−

37)

)a*1

"

"

1

et )*

) * "

6

6 " 8

3 " 1 8

)3 1*) 1* 8

1

)* 1

f p

f p

p p

p p

p p

p p

p

f

−

−

=

=− =

− − =

− − =+ − =

=

∴ =)$*

11

*")

"*")6

*")

"*)1

−=

=−

−−

=−−=−

k

k

k f

k f

38)

)a*

)$* 3*)8 ≤≤ x f

39)



( ) ( )

1313

1

8""

813

311

333

==+−

==−

=−−−−−

=+−−=−

=+−=

k k

h

h

hh

k h ) f(

k h ) f(

40)

38

6*1)"

1

1*")*1)

*")"3

3*3)*3)

*1)6"

"*")*")

3

3

=−=+−

−==− −

→−=+=++

→−=+=++

k

k

h

h

k h

k h

k h

k h

41)

)a*1

1

Aet )5*

o ) * 5

185

18 5 8

58

1 1"

"

1)5* 1"

"

f p

f p

p

p

p p

p

p

f

−

−

=

=+

=−

+ = −=

=

=

)$*

18

8*18*)1)

818!

18

18

*)

"

"

==−+

=−−

−=+

=−+

=

k

k k

k k

k k k

k k

k

k k f

42)

)a*



( )

[ ]

") 3* 3 6

15

3)15*

!

fg f

f

− = − + =

= +=

)$* 1Aet ) *

o ) *

3

3

f x y

f y x

y x

x y

− ==

+ =−

=

[ ]

1

1

"

) *

3

1 )1*

3 1

) 1* 6

7

x f x

gf g

g

−

−

−=

− =

= −

= − +=

43)

1 1) *

"

1

) *"

13

"1

""

5

"5

"

1

f k

f k

k

k

− =

=

−=

+

=

=

44)

)a*

"*")1 =− ff

)$*



1

1

et )*

o ) *

3 "

1

3 "

6

)* 6

f k

f k

k

k

k k

k

f

−

−

==

+=

−+ = −

=

∴ =

45)

)a*

"

18,

1816

1816*)

1816*)

"

"

−=

−=+=−=+=

−=++−=

q

qq p

q , pq p

xqq px p

x x f

)$*

5*1)

5

1"

1*)

*1)et

"*)

1

1

=∴

==−

==

−=

−

−

f

k

k

k f So

k f

x x f

46)

)a*

3

""7

7

"*7)

=

=−+

=

k

k

f

)$*1

1

Aet ) *

o, ) *

3

"

3

5 5

1

) * 1

f p

f p

p

p

p p

p

p

f

−

−

− == −

+= −

−+ = − +

==

∴ − =

47)

)a*1Aet ) * f x y

− =



o ) * f y x=

1

3

3

) *

3

y x

x y

x f x−

− =−

=

−=

)$*" ) * ) 3 *

3) 3 *

" !

! 16

f x f x

x

x

x

= −= − −= − += −

48)

)a*

" 1) *

31

1

33

f x f

x

x

x

= ÷

= ÷

=

)$*) * ) 3*

1

3) 3*

fg x f x

x

= −

=

−

49)

16*)8

16*5)

8*1)

1*)

*1)*) "

≤≤∴===

−=

x f

f

f

o f

x x f

50)

1,5

5618"

5186"

518*3)"

518*3)

518*)

==−=+−=

−=+−−=+−

−=−−=

h k

h ,k

xhkx

xhkx

xkx f

x x fg

51)



52)

53)

)a*1Aet )5* f p− =

[ ]

1

1

o ) * 5

" 3 5

1

)5* 1

)5* 1

1 13

1

"

f p

p

p

f

gf g

−

−

=+ =

=

=

=

+=

−= −

54)

)a*

) *

!

"

!

! "

7

f xkx

k

k

k

=

+=

++ =

= −

)$*

) 1*

7 1 !

3

f ( )

− =− − +

=

55)

)a*



" 1) *

1

11

1 1

11

1 1 1 )1 *

"

"

1

x f x f

x

x

x x

x

x x x x

x

x

+ = ÷− ++−=+

−−

− + +=− − +

=−

=−

)$* " "

"

) * ) *

1 ) *

1

1

f x f f x

f

x

x

x

=

=−

= − ÷−

=

56)

)a*

" 1) 1*

") 1* 1 1

f −

− =

− +=)$*

3 ") * ) *

" 1

" 1

1" 1

" 1 " 1

" 1

3 1

f x ff x

x f

x

x

x x

x

x

x x x

x

x

x

=

= ÷+

+=+

+

+=+ +

+

=+

57)

)a*

)* 3)* 1 11 f = −

=

)$*



( )1 1

1

1

1

3 )3 7*

)18*

Aet )18*

o ) * 18

3 1 18

11 3

11)3*

3

-

f g f

f

f k

f k

k

k

f g

− −

−

−

= +

=

==

− =

=

∴ =

58)

)a*

") "* ) 3*

"

)"*3)"*

"

1

"

gf g

g

−− = +

== −

= −

)$*

) * 1

3<) "* 1

3

3 1"

3

3 3"

fg x

f

x

x

x

x

= −

− = −

−

+ = −

−= −

− = −= −

59)

( )( )

q p

pq

p xq p xq

p xq p xq

p x g xq f

x gf x fg

−=−=

−−=+−+−=+−

+=−

=

3!"!6

*3)3"3"3

*3)3"

*)*)

60)

)a*

"

5

83

5"

=

=−−

x

x

x

chapter 1 functions paper 1 with answer

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