Fsc Part 2 Mathematics (Complete Solutions)

# Q1 The real valued function f and are defined below. Find

 (a)   ƒ o g (x) (b)   g o ƒ (x) (c)   ƒ o ƒ (x) (d)   g o g (x)
(i)             ƒ (x) = 2x + 1             ;             g (x) =
 3 x - 1
, x ≠ 1
(ii)             ƒ (x) = x + 1             ;             g (x) =
 1 x2
, x ≠ 0
(iii)             ƒ (x) =
 1 √x - 1
, x ≠ 1             ;             g (x) = (x2 + 1)2
(iv)             ƒ (x) = 3x4 - 2x2             ;             g (x) =
 2 √x
, x ≠ 0

Solution

(i)             ƒ (x) = 2x + 1             ;             g (x) =
 3 x - 1
, x ≠ 1
(a)   ƒ o g (x) = ƒ (g (x))
= ƒ (
 3 x - 1
)
= 2(
 3 x - 1
) + 1
=
 6 x - 1
+ 1
=
 6 + x - 1 x - 1
=
 5 + x x - 1
(b)   g o ƒ (x) = g (ƒ (x))
= g (2x + 1)
=
 3 2x + 1 - 1
=
 3 2x
(c)   ƒ o ƒ (x) = ƒ (ƒ (x))
= ƒ (2x + 1)
= 2(2x + 1) + 1
= 4x + 2 + 1
= 4x + 3
(d)   g o g (x) = g (g (x))
= g (
 3 x - 1
)
=
3
 3 x - 1
- 1
=
3
 4 - x x - 1
= 3 (
 x - 1 4 - x
)
=
 3 (x - 1) 4 - x

(ii)             ƒ (x) = x + 1             ;             g (x) =
 1 x2
, x ≠ 0
(a)   ƒ o g (x) = ƒ (g (x))
= ƒ (
 1 x2
)
=
 1 x2
+ 1
=
 1 + x2 x2
=
 √1 + x2 √x2
=
 √1 + x2 x
(b)   g o ƒ (x) = g (ƒ (x))
= g (x + 1)
=
 1 (√x + 1)2
=
 1 x + 1
(c)   ƒ o ƒ (x) = ƒ (ƒ (x))
= ƒ (x + 1)
=
 √x + 1 + 1
(d)   g o g (x) = g (g (x))
= g (
 1 x2
)
=
1
(
 1 x2
)2
=
1
 1 x4
= 1 (
 x4 1
)
= x4

(iii)             ƒ (x) =
 1 √x - 1
, x ≠ 1             ;             g (x) = (x2 + 1)2
(a)   ƒ o g (x) = ƒ (g (x))
= ƒ ((x2 + 1)2)
=
 1 √(x2 + 1)2 - 1
=
 1 √(x2)2 + 2(x2)(1) + 12 - 1
=
 1 √x4 + 2x2 + 1 - 1
=
 1 √x4 + 2x2
=
 1 √x2(x2 + 2)
=
 1 x √x2 + 2
(b)   g o ƒ (x) = g (ƒ (x))
= g (
 1 √x - 1
)
= ( (
 1 √x - 1
)2 + 1 )2
= (
 1 x - 1
+ 1 )2
= (
 1 + x - 1 x - 1
)2
= (
 x x - 1
)2
=
 x2 (x - 1)2
(c)   ƒ o ƒ (x) = ƒ (ƒ (x))
= ƒ (
 1 √x - 1
)
=
1
 1 √x - 1
- 1
=
1
 1 - √ x - 1
x - 1
=
1
 1 - √ x - 1
(x - 1)1/4
= 1 (
(x - 1)1/4
 1 - √ x - 1
)
=
(x - 1)1/4
 1 - √ x - 1
(d)   g o g (x) = g (g (x))
= g ((x2 + 1)2)
= (((x2 + 1)2)2 + 1)2
= ((x2 + 1)4 + 1)2

(iv)             ƒ (x) = 3x4 - 2x2             ;             g (x) =
 2 √x
, x ≠ 0
(a)   ƒ o g (x) = ƒ (g (x))
= ƒ (
 2 √x
)
= 3 (
 2 √x
)4 - 2 (
 2 √x
)2
= 3 (
 2 √x
)4 - 2 (
 2 √x
)2
= 3(
 (2)4 (√x)4
) - 2 (
 (2)2 (√x)2
)
= 3(
 16 x2
) - 2 (
 4 x
)
=
 48 x2
-
 8 x
=
 48 - 8x x2
=
 8(6 - x) x2
(b)   g o ƒ (x) = g (ƒ (x))
= g (3x4 - 2x2)
=
 2 √3x4 - 2x2
=
 2 √x2(3x2 - 2)
=
 2 x2√3x2 - 2
(c)   ƒ o ƒ (x) = ƒ (ƒ (x))
= ƒ (3x4 - 2x2)
= 3(3x4 - 2x2)4 - 2(3x4 - 2x2)2
(d)   g o g (x) = g (g (x))
= g (
 2 √x
)
=
2
 2 √x
=
2
 √x 2
=
2√
 √ x
2
Divide and multiply by 2
= (
2√
 √ x
2
) (
 √2 √2
)
=
22
 √ x
22
=
22
 √ x
(2)2
=
22
 √ x
2
=
2
 √ x