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
x23x2 - 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

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