functions and inverse functions · mathematics grade 12 inverse functions 17 3.1.2 graph of an...

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


2

2.3 ( ) (

)

4

2.4 ( ) √

3

2.5 ( )

4


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 ( )

( )


4

1.2 ( )

( )

1.3 ( )

√

( ) √

1.4 ( )

√

( ) √

1.5 ( )

( ) ( ) ( )

1.6 ( ) (

)

(

)

(

)

( )

( ) ( )

1.7 ( )

( )

( ) ( )

( )

( ) ( ) ( ) ( ) ( )

( )


5

1.8 ( )

( )

(16)

2. (4 marks each function/4 marks each inverse)

2.1

𝑓

𝑓


6

2.2

2.3

𝑔

𝑔


7

2.4

2.5

k

𝑘

𝑗

𝑗


8

2.6

2.7

𝑙

𝑙

𝑚

𝑚


9

2.8

(64)

3 (2 marks each function/ 2 marks each inverse)

Function Inverse

3.1 Domain: Range: ℝ ℝ

Domain: Range: ℝ ℝ



3.3 Domain: Range: ℝ

Domain: Range: ℝ


Domain: Range: ℝ


Domain: Range: ℝ


Domain: Range: ℝ


Domain: Range: ℝ



(32) TOTAL: 112

𝑛

𝑛


10

ACTIVITY 3 ( 50 marks)

QUESTION 1

( ) ( ) and ( ) ( )

1.1

(3)

1.2 ( )

( )

(5)

1.3

(9)

[17]

𝑓

𝑔


11

QUESTION 2

Given: ( )

2.1

; (2)

2.2 (2)

2.3

(2)

2.4 . (1)

[7]


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)


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)


14

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


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


15

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


16

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)


17

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


18

3.2.3 )2(5)( xxf Exponential graph

f(x) = 5x

x substituted by (x + 2) (2)

50


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)


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]


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

𝑓

functions and inverse functions · mathematics grade 12 inverse functions 17 3.1.2 graph of an...

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