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MATHEMATICS GRADE 12 INVERSE FUNCTIONS
1
FUNCTIONS AND INVERSE FUNCTIONS
SOLUTIONS ACTIVITY 1 (50 marks
1.
1.1 One – to – one function 1.2 One – to – many relation
1.3 Many – to – one function 1.4 Many – to – one function
1.5 One – to – one function
1.6 Not a function/
One – to – many relation
1.7 One – to – many relation
1.8 One – to – one function
1.9 Many – to – one function
1.10 Not a function/
One – to – many relation
(10)
2.1 ( )
2
2.2 ( )
4
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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2.3 ( ) (
)
4
2.4 ( ) √
3
2.5 ( )
4
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
3
3.
( ) DOMAIN RANGE ℝ ℝ
( ) DOMAIN RANGE ℝ
( ) (
)
DOMAIN RANGE
ℝ
( ) √ DOMAIN RANGE
( )
DOMAIN RANGE
ℝ ; y ℝ ;
(10)
4.
( ) ( )
( )
( ) ( )
( ) ( ) ( )
( ) (
)
( ) √
( )
(10)
5. The vertical line cuts it once, every - value is associated with only one - value. (3) TOTAL: 50
ACTIVITY 2 (112 marks)
1.1 ( )
( )
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
4
1.2 ( )
( )
1.3 ( )
√
( ) √
1.4 ( )
√
( ) √
1.5 ( )
( ) ( ) ( )
1.6 ( ) (
)
(
)
(
)
( )
( ) ( )
1.7 ( )
( )
( ) ( )
( )
( ) ( ) ( ) ( ) ( )
( )
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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1.8 ( )
( )
(16)
2. (4 marks each function/4 marks each inverse)
2.1
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MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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2.2
2.3
𝑔
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MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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2.4
2.5
k
𝑘
𝑗
𝑗
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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2.6
2.7
𝑙
𝑙
𝑚
𝑚
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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2.8
(64)
3 (2 marks each function/ 2 marks each inverse)
Function Inverse
3.1 Domain: Range: ℝ ℝ
Domain: Range: ℝ ℝ
3.2 Domain: Range: ℝ ℝ
Domain: Range: ℝ ℝ
3.3 Domain: Range: ℝ
Domain: Range: ℝ
3.4 Domain: Range: ℝ
Domain: Range: ℝ
3.5 Domain: Range: ℝ
Domain: Range: ℝ
3.6 Domain: Range: ℝ
Domain: Range: ℝ
3.7 Domain: Range: ℝ
Domain: Range: ℝ
3.8 Domain: Range: ℝ ℝ
Domain: Range: ℝ ℝ
(32) TOTAL: 112
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𝑛
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
10
ACTIVITY 3 ( 50 marks)
QUESTION 1
( ) ( ) and ( ) ( )
1.1
(3)
1.2 ( )
( )
(5)
1.3
(9)
[17]
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MATHEMATICS GRADE 12 INVERSE FUNCTIONS
11
QUESTION 2
Given: ( )
2.1
; (2)
2.2 (2)
2.3
(2)
2.4 . (1)
[7]
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
12
QUESTION 3
Given: ( )
for
3.1
4
3.2 ( )
– , (3)
[7]
QUESTION 4
4.1 ( ) and ( )
4.1.1 ( )
Turning point:
( )
( )
OR f’(x) = 4x – 6
0 = 4x – 6
x =
(
) (
) (
)
D (
) OR D (
) (2)
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
13
4.1.2
( )( )
or
A( ) B( ) (3)
4.1.3 ( )
Substitute ( ):
( )
(2)
4.1.4
OR
For non-real roots
( ) ( )( )
4.1.5 ( ) ( )
or (2)
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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4.1.6
Tangent if roots are equal:
( ) ( )( )
(4)
4.2 ( ) and ( ) .
4.2.1 ( ) and ( )
( ) ( )
( )
Shift p 1 unit to the left and 2 units downwards (3)
4.2.2 ) OR (1)
[19]
TOTAL: 50
ACTIVITY 4 (50 marks)
QUESTION 1 [14]
1.1 ( )a
f x qx p
If the asymptotes intersect at (–1 ; 8) then p = 1 and q = 8.
12 82 1
a
12 81
a
12 8a
4 a
p = 1
q = 8
Substitution
a = -4
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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1( )
2
x
g x b
21
122
b
12 .4b
12 4 3b
Substitution
b = 3
The hyperbola 4
yx
has shifted 1 unit to the left and 8 units
upwards.
The line of symmetry will do the same:
So: y x becomes ( ) 1( 1) 8k x x
( ) 9k x x
m = 1 and c = 9
Equation
m = 1
c = 9
(9)
1.2
13
( ) 2
3 3
x
g x
1
2
x
y
1
2
y
x
1
2
logy x
13
( ) 2
3 3
x
g x
Interchanging x and y
Equation
(3)
1.3
Shape
x - intercept
(2)
QUESTION 2 [22]
2.1.1 RK = 3 units
Answer (1)
2.1.2 KM = 5 units Answer (1)
2.2 1 < x < 3 1 < x
x < 3 (2)
2.3 0 < x < 1 or x > 3
0 < x
x < 1
x > 3 (3)
x
0
(1 ; 0)
y
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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2.4 x at H = –5 5
12,5
2OH
32 2,5OH
34OH units
x = –5
Substitution
32
34units (4)
2.5 As x increases, y decreases
OR
2 1 2 1, .If x x then y y
OR
The gradient at each point on the graph is negative.
as x increases (2)
y decreases
OR
2 1If x x
2 1.then y y (2)
OR
Gradient at each point
Negative (2)
2.6 ( 3) 5y x becomes y x
2y x
1x
+5 (2)
2.7 , 3x R x
x R
3x (2)
2.8 1 14 (2) 0,625 ( 3)x x
1 1 625 1
4 2 1 000 ( 3)
x
x
2 5001 1
2 1 000 ( 3)
x
x
1 42,5
2 ( 3)
x
x
1 42,5
2 ( 3)
x
x
Answer: at P(1 ; 3) because the question actually asks
where do the two graphs intersect.
1
( 3)x
1
4
2 500
1 000
2,5
Reason
(5)
QUESTION 3 [14]
3.1.1 0x or 0x 0x
0x (2)
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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3.1.2
Graph of an inverse
relation of a parabola
Correct position of the
graph in the 2nd
and 3rd
quadrants.
(2)
3.1.3 It does not pass the vertical line test
OR
It is a many-to-one mapping.
Answer
OR
Answer (2)
3.2.1
x
axf )(1
x
ay
252
a
Square both sides:
254
a
a100
xxg blog)(1
xy blog
25log2 b
252 b 22 5b
5b
Substitution
Squaring both sides
Answer
Substitution
Answer
(5)
3.2.2 250 x
OR ]25;0(x
x0
25x
OR
(0
25] (2)
x
y
0
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
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3.2.3 )2(5)( xxf Exponential graph
f(x) = 5x
x substituted by (x + 2) (2)
50
ACTIVITY 5 (51 marks)
QUESTION 1
1.1 ( ) (2)
1.2
( )
√
( √ )
(3)
1.3
( )
( )
(3)
1.4
(3)
1.5
(3)
1.6 √ √ (4)
[18]
QUESTION 2
2.1 Yes For every x-value there is only one corresponding y-value
(2)
2.2 ( ) (1)
2.3 ( )
(2)
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
19
2.4 ( )
For
√
√
(3)
[8]
QUESTION 3
3.1.1 ( ) ( ) ( )
(4)
3.1.2 ( ) ( )
(3)
3.2 2 units left; 5 units up (2)
3.3
( ) ( )
(5)
[14]
MATHEMATICS GRADE 12 INVERSE FUNCTIONS
20
QUESTION 4
4.1 ( )
(1)
4.2
(1)
4.3 For
(2)
4.4
(3)
4.5
(2)
4.6 Reflected through the ; shifted down 2 units
(2)
[11]
Total: 51
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