Raffles Junior College
H2 Mathematics 9740
JC1 2008
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Assignment 3
Functions, Inverse functions and Composite functions
[MJC06H2/CT/Q8]
1 Functions f and g are defined by
f :x
g: x
 x  3  6 ,
ln  x  2 ,
2
x  R, x  2,
x  R, x  3 .
(i) Show, by means of a graphical argument or otherwise, that f 1 exists, and
define f 1 in a similar form.
(ii) Prove that the composite function gf exists and find an expression for gf  x  .
Find the range of gf.
2
[6]
[4]
The functions f and g are defined as follows:
f:x
cos x, 0  x   ,
g: x
tan x, 

2
x

2
(i) State the range of each of the functions f and g.
(ii) Find f 1 and g 1 , and state the range of each of the inverse functions.
[2]
[4]
(iii) Sketch, on separate diagrams, the graphs of f 1 and g 1 .
[4]
[ACJC06H2/Promo/Q13]
3 A function f is defined by f : x  ( x  a)( x  b), x  , x  b , where 0  a  b .
Sketch the graph of f and find its range.
Another function g is defined by g : x  e x , x 
[2]
.
Show that g 1f exists and define g 1f in similar form. Find also the range of g 1f .
[7]
[TJC06H1/CT/Q8 modified]
4 Functions f and g are defined as follows:
f : x  x 2 , x  , x  3
g : x  x, x  , x  0
1
(i) Define f in a similar form.
[2]
(ii) Sketch, on a single diagram, the graphs of y  f  x  , y  f 1  x  and y  ff 1  x  ,
showing clearly the relationships between the three graphs.
(iii) Explain why gf exists. Define gf in a similar form.
[3]
[3]
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Assignment 3: Functions, Inverse functions and Composite functions
Page 1 of 6
Raffles Junior College H2 Mathematics (9740) JC1 2008
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5
The functions f and g are defined by
1
x
e , x  R, x  0
x2 , x  R .
f: x
g: x
(i) Sketch the graph y  f ( x) and determine the range of f.
[3]
(ii) Determine which of the composite functions fg and gf exist.
[4]
(iii) Show by means of a graphical argument or otherwise, that f is one-one.
[1]
(iv) Find f 1 ( x) and give its domain and range.
[3]
[VJC06H2/Promo/Q14]
6 The function f is defined by
3   x  1 , x  2.
2
f:x
(i) Find the range of f.
(ii) State, giving a reason, whether ff exists.
The function g is defined by
g: x
f  x ,
[1]
[2]
x  1.
(iii) Sketch the graphs of y  g( x) , y  g 1 ( x) and y  g 1g( x) on a single diagram,
showing clearly their geometrical relationships.
[4]
1
(iv) If g( )  g ( ) , find the values of the constants p and q such that
 2  p  q  0 .
[2]
(v) Find an expression for g 1 ( x) .
[3]
[NJC07H1/CT/Q2]
7 The functions f , g and h are defined as follows:
f :x
g:x
h:x
x 2  4 x  1, x  k
1
1 ,
x
2x
e ,
x
x
(i) State the smallest value of k such that f 1 exists and express f 1 in a similar form. [3]
(ii) Express each of the following functions in terms of f, g and h as appropriate.
(a) x
ln x , x  0
(b) x
1
1
, x
e2 x
(c) x
x,
x  3
[4]
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Assignment 3: Functions, Inverse functions and Composite functions
Page 2 of 6
Raffles Junior College H2 Mathematics (9740) JC1 2008
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Worked Solutions:
1(i) y = f(x) y
Since any horizontal line y  b, b  7 , cuts the
graph of y  f  x  at one and only one point, f is
y=b
15
one-one. Hence, f 1 exists.
(2, 7)
x
0
Let
y  ( x  3) 2  6
( x  3) 2  y  6
x  3 y 6
Since x  2, x  3  y  6
Df -1  R f  [7, )
3  x  6,
f -1 : x
1(ii)
x7
Since R f  [7, )  (3, )  Dg ,  gf exists.
gf ( x)  g  ( x  3) 2  6 
 ln  ( x  3) 2  6  2 
y
 ln  ( x  3) 2  4 
y = g(x)
R gf  [ln 5, )
0
2(i)
x
3
y
y
y  f  x
1

2
1
R f   1, 1
2(ii)
y  g  x

x
x
0
x   2
x

2
Rg 
Let y  cos x  x  cos 1 y .  f 1  x   cos1 x, x  1, 1 . R f 1  0,  
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Assignment 3: Functions, Inverse functions and Composite functions
Page 3 of 6
Raffles Junior College H2 Mathematics (9740) JC1 2008
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Let y  tan x  x  tan 1 y .  g1  x   tan 1 x, x 
y
2(iii)
  
. R g 1    , 
 2 2
y

y
y  f 1  x 

yg
2
1
3
0
f(x)  ( x  a)( x  b),
 x
2
x
y   2
x
1
1

xb
y
y  f  x
x
b
From the graph, Rf is (0, )
g is a one-one function  g 1 exists.
D 1  R g  (0, )
g
Since R  D
f
g1
,  g 1f exists.
g(x)  e x , x  .
Let y  g( x)  y  e x  x  ln x
 g 1( x)  ln x, x  0
g 1f ( x)  ln[( x  a)( x  b)], x  b
R 1  R 1 when x  R f
g f
g
R
4(i)
g 1
when x  (0, ) 
Let y  x2  x   y or
Hence f 1 : x
y (rejected since x   3 )
 x, x  , x  9 .
y
y = x2
9
-3
x
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Assignment 3: Functions, Inverse functions and Composite functions
Page 4 of 6
Raffles Junior College H2 Mathematics (9740) JC1 2008
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y
4(ii)
y = f(x)
y  ff 1  x 
 9, 9 
 3, 9 
x
 9,  3
y  f 1  x 
y=x
4(iii) Since Rf = (9, )  (0, ) = Dg, thus gf exists.
Dgf = Df = (–, 3).
gf ( x)  x 2  | x |
 gf : x
 x, x  , x  3
5(i)
y  f  x
y  f  x
y 1
Rf 

\ 1
1
2
5(ii)
Since R g 0,   
Since R f 

\ 0  Df
\ 1 
, so fg does not exist.
 Dg , so gf exists.
5(iii) From the sketch of y  f  x  , any horizontal line y  k , k 

\ 1 , cuts the graph
of y  f  x  at one and only one point. Thus, f is one-one.
1
5(iv)
Let y  e x  x 
f 1  x  
6(i)
6(ii)
1
, x
ln x
1
.
ln y

\ 1 ; R f 1  \ 0
R f   , 3
Since R f   , 3   , 2  Df , so ff does not exist.
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Assignment 3: Functions, Inverse functions and Composite functions
Page 5 of 6
y
y
= g(x)
Raffles Junior College H2 Mathematics (9740) JC1 2008
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y
6(iii)
(1, 3)
0
y=x
2
3
x
(1, 1)
(3, 1)
1 3
2
y  g1  x 
1 3
y  g 1g  x 
6(iv)
x
y  g  x
Since g    g1  
g    
3    1  
2
2    2  0
Hence, p   1 and q   2
6(v)
Let y  3   x  1
2
x  1  3 y
Since x  1 , x  1  3  y ,  g 1  x   1  3  x , x  3
7(i)
k  2
y
y = f(x)
1
0.268
3.73
x
(2, 3)
Let y  x 2  4 x  1
y  x  2  3
2
y  3  x  2
2
 x  2 
y  3 or  y  3
x  y  3  2 or  y  3  2
Since x  2, x  y  3  2
 f 1 : x
x  3  2, x  3
7(ii) (a) h 1 ( x)  ln x , x  0
(b) gh( x)  1 
1
, x
e2 x
(c) ff 1 ( x)  x, x  3
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Assignment 3: Functions, Inverse functions and Composite functions
Page 6 of 6