02 Functions - IGCSE PDF

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0606 Additional Mathematics

Unit 02 Functions

2 Mathematical Formulae

y m e d a n c i v A l n A i y y lv m m A e e d y d a m ( ) ( ) ca ( ) Ac e A d n i ca ( ) vin A lv l A n A i y y lv m m A e e d d a a c c A A n i lv A 1.  ALGEBRA

Quadratic Equation

For the equation ax2 + bx + c = 0,

x=

−b

b 2 − 4 ac 2a

Binomial Theorem

n n n (a + b)n = an + 1 an–1 b + 2 an–2 b2 + … + r an–r br + … + bn,

n n! where n is a positive integer and r = (n – r)!r!

2.  TRIGONOMETRY

Identities

sin2 A + cos2 A = 1

sec2 A = 1 + tan2 A

cosec2 A = 1 + cot2 A

Formulae for ∆ABC

a c b = = sin A sin B sin C a2 = b2 + c2 – 2bc cos A ∆=

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1 bc sin A 2

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0606 Additional Mathematics

Unit 02 Functions

14

0606/21/M/J/14 12  The functions f and g are defined by

2x f ^x h = + for x 2 0 , x 1

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A g ^x h = x + 1 for x 2-1.

 

(i)  Find fg^ 8h.

[2]

 

ax (ii)  Find an expression for f 2 ^x h, giving your answer in the form , where a, b and c are integers bx + c to be found. [3]

 

(iii)  Find an expression forg -1 ^ xh, stating its domain and range.

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[4]

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0606 Additional Mathematics

Unit 02 Functions

15

0606/21/M/J/14 12 

(iv)  On the same axes, sketch the graphs of y = g ^x h and y = g 1^ xh, indicating the geometrical relationship between the graphs. [3]

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A y

O

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x

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0606 Additional Mathematics

0606/22/M/J/14

Unit 02 Functions

12

11  The functions f and g are defined, for real values of x greater than 2, by f( x) = 2 x - 1,

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A g (x ) = x ^x + 1h.

 

 

(i)  State the range of f.

[1]

(ii)  Find an expression for f -1 (x) , stating its domain and range.

[4]

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0606 Additional Mathematics

0606/22/M/J/14 11 

Unit 02 Functions

13

(iii)  Find an expression forgf (x )and explain why the equation gf (x ) = 0 has no solutions.

[4]

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A

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0606 Additional Mathematics

Unit 02 Functions

16

0606/23/M/J/14 12  The function f is such that f ( x) = 2 +

x - 3 for 4 G x G 28 .

(i)  Find the range of f.

[2]

 

(ii)  Find f 2 (12) .

[2]

 

(iii)  Find an expression for f -1 (x) .

[2]

 

The function g is defined by g (x) = 120 for x H 0. x

 

(iv)  Find the value of x for whichgf (x )= 20 .

 

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A

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[3]

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0606 Additional Mathematics

Unit 02 Functions

5

0606/21/O/N/14 4 

The functions f and g are defined for real values of x by f ( x) =

x- 1 - 3

for x 2 1,

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A g (x ) =

x-2 2x - 3

for x 2 2.

(i)  Find gf(37).

[2]

 

(ii)  Find an expression for f - 1 (x) .

[2]

 

(iii)  Find an expression for g -1 (x).

[2]

 

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0606 Additional Mathematics

0606/23/O/N/14

Unit 02 Functions

10

The functions f and g are defined for real values of x by

7 

2 f ^x h = x + 1 for x 2 1,

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A g^x h = x 2 + 2 .

Find an expression for

   

 

 

(i) f -1 ^x h,

[2]

(ii) gf ^ xh,

[2]

(iii)  fg ^ xh.

[2]

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0606 Additional Mathematics

0606/23/O/N/14 7 

Unit 02 Functions

11

+ (iv)  Show thatff ^ xh = 3x 2 and solve ff^ xh = x. x +2

[4]

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A

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0606 Additional Mathematics

Unit 02 Functions

5

0606/12/F/M/15 3 

(i)  On the axes below sketch the graph of y = 4 - 5x , where the graph meets the coordinate axes.

stating the coordinates of the points [3]

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A y

O

 

(ii)  Solve

4 - 5x = 9 .

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x

[3]

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0606 Additional Mathematics

Unit 02 Functions

10

0606/12/F/M/15

f (i) = sin 2i for 0 G i G

r

.

8 

(a)  A function f is such that

 

 

(i)  Write down the range of f.

[1]

 

 

(ii)  Write down a suitable restricted domain for f such that f -1 exists.

[1]

 

2

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A

(b)  Functions g and h are such that

g (x ) = 2 + 4 lnx for x 2 0 , h (x) = x 2 + 4 for x 2 0 .

 

 

(i)  Find g -1 , stating its domain and its range.

 

 

(ii)  Solve

( )gh x = 10 .

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[4]

[3]

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0606 Additional Mathematics

Unit 02 Functions

12

0606/22/M/J/15

10  (a)  The function f is defined by f : x 7 sin x graph of y = f (x ).

for 0° G x G 360°. On the axes below, sketch the [2]

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A y

O

 

90°

180°

270°

360°

x

(b)  The functions g and hg are defined, for x H 1, by

g ^x h = ln^4x - 3h, hg ^ xh = x.

 

 

(i)  Show that

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e h^ xh4=

x

+3 .

[2]

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0606 Additional Mathematics

Unit 02 Functions

13

0606/22/M/J/15  

 

(ii) y

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A y = g(x)

O

1

x

The diagram shows the graph of y = g^ xh . Given that g and h are inverse functions, sketch, on the same diagram, the graph of y = h ^x h. Give the coordinates of any point where your graph meets the coordinate axes. [2]

 

 

 

 

(iii)  State the domain of h.

[1]

 

 

(iv)  State the range of h.

[1]

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0606 Additional Mathematics

10

0606/13/M/J/15 8   

It is given that      

Unit 02 Functions

f^ xh = 3e2x for x H 0, x H 0. g ^x h = ^x + 2h2 +      for 5

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A (i)  Write down the range of f and of g.

[2]

 

(ii)  Find g-1 , stating its domain.

[3]

 

(iii)  Find the exact solution of gf^ xh = 41.

[4]

 

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0606 Additional Mathematics

0606/13/M/J/15 8 

Unit 02 Functions

11

(iv)  Evaluatef l^ln 4 h.

[2]

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A

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0606 Additional Mathematics

Unit 02 Functions

4

0606/23/M/J/15 2 

(a) y

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A 4

2

O

 

x

 

The diagram shows the graph of fy( =) x passing through ^ 0, 4h and touching the x-axis at ^2, 0h. Given that the graph of y = f (x )is a straight line, write down the two possible expressions for f (x ). [2]

 

(b)  On the axes below, sketch the graph of y = e-x + 3, stating the coordinates of any point of intersection with the coordinate axes. [3] y

O

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x

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0606 Additional Mathematics

Unit 02 Functions

14

0606/12/O/N/15

11   (a)  A function f is such that f^ xh = x2 + 6x + 4 for x H 0 .

x 2 + 6x + 4 can be written in the form ^ x + ah2 + b, where a and b are integers. [2]

 

 

(i)  Show that

 

 

(ii)  Write down the range of f.

[1]

 

 

(iii)  Find f - 1 and state its domain.

[3]

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A

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0606 Additional Mathematics

Unit 02 Functions

15

0606/12/O/N/15 11 

(b)  Functions g and h are such that, for x d R ,

g ^x h = e x

 

 

and

h ^x h = 5x + 2.

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A

Solve h g x

2

^37h =

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.

[4]

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0606 Additional Mathematics

Unit 02 Functions

8

0606/13/O/N/15 6 

y = x2 - 4x- 12

(i)   On the axes below, sketch the graph of pointswherethegraphmeetstheaxes.

showing the coordinates of the [3]

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A y

O

 

 

x

(ii)  Findthecoordinatesofthestationarypointonthecurve

(iii)  Findthevaluesofksuchthattheequation

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y=

x 2 - 4x - 12 = k

2

x - 4x- 12.

[2]

hasonly2solutions.

[2]

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0606 Additional Mathematics

0606/23/O/N/15

Unit 02 Functions

11

9 

Given that f (x) = 3x 2 + 12x + 2,

 

(i)  find values of a, b and c such that f (x)= a (x+ b)2 + c,

y m e d a n c i v A l n A i y y lv m m A e e d y d a c m ca e A A d n n i ca i A lv lv A n A i y y lv m m A e e d d a a c c A A n i lv A

[3]

 

(ii)  state the minimum value of f(x) and the value of x at which it occurs,

[2]

 

1 (iii)  solve fc m = 0 , giving each answer for y correct to 2 decimal places. y

[3]

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