Composite Functions

EXERCISE 1.3(a)

1. The functions of f and g are defined as f(x)=2x and g(x) = 3 – x . Find

(a) the composite function of g f and the value of g f (4).

g f(x) =

Let f(x) = ySolution,

g (y)

g (x) = 3 – x

g (y) = 3 - y

= 3 - y

= 3 – 2x

g f (x) = 3 – 2x

g f (4) = 3 – 2(4)

= 3 – 8

= - 5

1. The functions of f and g are defined as f(x)=2x and g(x) = 3 – x . Find

(b) The composite function of f g and the value of f g (4).

f g(x) =

Let g(x) = y

Solution,

f (y)

f (x) = 2 x

f (y) = 2 y

= 2 y

= 2 (3 – x) = 6 – 2x

fg(x)= 6 – 2x

fg(4)= 6 – 2(4)

= 6 – 8

= -2

1. The functions of f and g are defined as f(x)=2x and g(x) = 3 – x . Find

Solution,

(c) f f (2)

f f (2)

f (x) = 2x f (2) = 2(2) = 4

= f (4) f(x) = 2xf (4) = 2(4) = 8

= 8

1. The functions of f and g are defined as f(x)=2x and g(x) = 3 – x . Find

Solution,

(d) gg ( - 4 )

gg (-4)

g(x) = 3 - x g(-4) = 3 – (-4) = 3 + 4 = 7

= g (7)

g (7) = 3 - 7 = -4

= -4

2. Function f and g are defined by f:x|→3x + 1, g:x|→1 - x² . obtain expressions in similar form for

(a) g f (x)

g f(x) = g (y) g (x) = 1 - x²

g (y) = 1 - y²

Let f (x) = y

= 1 - y²

= 1 – (3x+1)²

= 1 – (3x+1)(3x+1)

= 1 – (9x²+3x+3x+1)

= 1 – (9x²+6x+1)

= 1 – 9x² - 6x - 1

= – 9x² - 6x

2. Function f and g are defined by f:x|→3x + 1, g:x|→1 - x² . obtain expressions in similar form for

(b) f g(x)

fg(x) = f (y)

f (x) = 3x + 1

f (y) = 3y + 1

Let g(x) = y

= 3y + 1

= 3(1-x²) +1

= 3 - 3x² +1

f ² (x)

= 4 - 3x²

2. Function f and g are defined by f:x|→3x + 1, g:x|→1 - x² . obtain expressions in similar form for

ff(x) = f (y)

f (x) = 3x + 1

f (y) = 3y + 1

Let f(x) = y

= 3y + 1

= 3(3x + 1) + 1

= 9x + 3 + 1

(c) f ² (x)

= 9x + 4

from (b)

(d) Find the value of x for which fg(x) = 1.

fg(x) = 4 - 3x²

4 - 3x² = 1

4 - 1= 3x²

3 = 3x²

3x² = 3

x² = 1

1,1 x

3

)(xff

0,3

)( xx

xg

4)4( x

)(yf

8x

4y

4)( xxf

(a) f ² let f(x)=y = x - 4

3

0,3

)( xx

xg

x

4)( xxf

4x

82 xf

44 x

4)( xxf

(a) f ²4)( xxf

3

0,3

)( xx

xg

x

4)( xxf

4x

4

3)(

xxgf

4

3

x

4)( xxf

(b) g f

xxg

3)(

4

kxx

xxg

,2

1)(

02 x

2k

2x

13)( xxf

(a) Determine the value of k.

kx 2x

4

kxx

xxg

,2

1)(

x2

1

x

x

12

33)(

x

xxfg

1)2

1(3

x

x

13)( xxf

(b) f g

13)( xxf2

1)(

x

xxg

4

kxx

xxg

,2

1)(

x 13 x

13

23)(

x

xxgf

2)13(

1)13(

x

x

13)( xxf

(c) gf

13)( xxf2

1)(

x

xxg

5

xxg 21)(

x x21

2)63()( xxfg

2)21(3 x

23)( xxf

(a) f g (x)

23)( xxfxxg 21)(

x65

5

xxg 21)(

x 23 x

)46(1)( xxgf

)23(21 x

23)( xxf

(b) gf(x)

23)( xxf xxg 21)(

461 x

36 x

5

xxg 21)(

2123 xx

23)( xxf

(c) Find the value of x for f(x) = g(x)

)()( xgxf

xx 2123

15 x

5

1x

5 xxg 21)(

22 x

23)( xxf

(d) Find the value of x if f(x) maps to itself.

xx 23

23 xx

12

2x

6 5)( xxg

1 6

116 ba

ba 6

baxxf )(

(a) Calculate the value of a and b.baxxf )(

fg(1)=11 , gf(2)=8

5)( xxg

2 ba 2

852 ba

52 ba

5)( xxgbaxxf )(

32 ba

1

2

(1) – (2)

116 ba

32 ba

84 a2a

From equation (2) 32 ba

3)2(2 b

34 b1b

6 5)( xxgbaxxf )(

(b) What are the value of gf(3)?

12)( xxf

10

)3(gf

1)3(2)3( f

1,2 ba

)5(g

from (a)

5)3(f

55

7 32)( 2 xxxh

)(xhg

15)( xxg

Find hg(x).yxg

let)( 15 x

)(yh

322 yy

3)15(2)15( 2 xx

)()(2 xhhxh )(yh

bay bbaxa )(

baxyxh )(

babxa )( 2

babxa 2

By comparing,

362 a6a

35bab356 bb357 b5b

)(xhg )(yh

342 yy

3)25(4)25( 2 xx

9.

Let g(x) = y = 5x+2

10

(a) f(5)= 5 - 1

= 4

(b) gf(5)=14

g(4) = 14k(4)+2 =

14 4k = 14 - 2

4k = 12k = 12÷4k = 3