Functions
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
1
Calculus I
Chapter 1
Functions
1.1
Introduction
2
1.2
Direct Images and Inverse Images
4
1.3
Composition of Functions
5
1.4
Constant Function and Identity Function
6
1.5
Injective, Surjective and Bijective Functions
7
1.6
Some Special Real Functions
12
1.7
Elementary Functions
21
1.8
Revision Exercise
22
Hung Fung Book Calculus and Analytical Geometry I
Functions Revision Exercise P.49 ( 1- 15 )
Prepared by K. F. Ngai
Page 1
Functions
1.1
Advanced Level Pure Mathematics
Introduction
Given a set A which has two elements x, y .
We denote A  x, y or A  y, x . We can write A either equals x, yor y, x. It is unordered pair.
In coordinate system, the x and y coordinates are written in ( x, y )  ( y, x) . It is ordered pair.
It is easy to see that two ordered pairs ( x1 , y1 ) and ( x 2 , y 2 ) are equal if and only if x1  x 2 and y1  y 2 .
Definition
A function (or a mapping ) from a set A into a set B , is defined as f : A B
(i) Pr1 f  A
(ii) (a1 , b1 ), (a2 , b2 )  f , if a1  a 2 , then b1  b2 .
For any a  A , f (a ) is unique.
f (a )  value of f at a .
Pr1 f is the first projection of the ordered pair of f .
Pr2 f  Image of f
A  Domain of f
B  Range of f
Example
Let A  1,2,3,4,5, B  a, b, c, d , e
The following are functions (mappings) from A to B .
f1  (1, a), (2, c), (3, b), (4, c)
f 2  (1, d ), (2, c), (3, b), (4, a)
The following are not functions (mappings) from A to B .
g1  (1, a), (2, b), (3, c)
g2  (1, a), (1, b), (2, a), (3, a), (4, c)
g 3  (1, d ), (2, e), (3, f ), (4, h)
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Functions
Advanced Level Pure Mathematics
Remark
A function of real variable is a function whose domain is the set of all real numbers or a subset
of R . A real-valued function is a function whose range is the set of all real numbers.
Example
Let x  R , find the domain as long as possible of each of the following functions.
(a)
f ( x)  2 x  5
(b)
f ( x) 
(c)
f ( x)  x
(d)
f ( x) 
(e)
f ( x)  1  x
(f)
f ( x) 
(g)
f ( x )  tan x
(h)
f ( x) 
(i)
f ( x )  log( x  1)
(j)
f ( x )  sin x
1
x
x
x  2x  3
2
1
sin x
x
1
sin x
x , x  0 , then f : [0,)  [0,)
Example
If f ( x ) 
Example
In the following, which is/are graph(s) of a function(s) of x ?
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Functions
Example
Advanced Level Pure Mathematics
For each of the following pairs of functions, are they identical ? If no, explain.
x
, g( x)  1
x
(a)
f ( x) 
(b)
f ( x)  x , g( x) 
(c)
(d)
f ( x )  1 , g ( x )  sin 2 x  cos 2 x
x2
f ( x )  ln x 2 and g ( x )  2 ln x
1.2
Direct Images and Inverse Images
Definition
Let f : A  B be a function from A to B and X  A .
 f X 
f X    f (a) : a  X 
The direct image ( image ) of X under f
Example
is defined as
(1) Let the function f : A  B be represented by the following figure.
If X  2,3,4 , then f X    f (2), f (3), f (4)  b, d , e.
(2)
g : 0,2   C
g ( x )  cos x  i sin x , where i 2  1 .
If X  0,   , then gX  is a unit semi-circle above the real axis.
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Functions
Definition
Advanced Level Pure Mathematics
Let f : A  B be a function from A to B and Y  B .


The Inverse image f 1 Y  of Y under f is defined as
f 1 Y   x : x  A and f ( x)  f 
Example
(1) Let the function f : A  B be represented by the following figure.
If Y  2,3,4, then f 1[Y ]  a, b, c, d  .
(2) Let f : R  R . f ( x )  sin x .
f 1 0 
1.3
Composition of Functions
Definition
Let f : A  B and g : B  C be two functions. The composition of f and g ,
denoted by g  f , is a function from A to C such that a  A , g  f (a )  c with
f (a )  b and g (b)  c , where b  B.
i.e. ( g  f )(a)  g f (a) , a  A .
Example
Let A  1,2,3,4,5, B  x, y, z, w, C  a, b, c, d , e, f .
f  (1, x), (2, y), (3, z), (4, z), (5, x)
g  ( x, a), ( y, d ), ( z, e), (w, a)
g  f  (1, a), (2, d ), (3, e), (4, e), (5, a)
Example
If f ( x )  sin x , x  R , g ( x )  1  x , x  (,1] ,
then g  f  1  sin x , x  R .
f : R   1,1, g : (,1]  0, 
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Functions
Advanced Level Pure Mathematics
Given that f ( x )  2 x  1 , g ( x )  x 2  1 x  R
Example
Find
f  g (0)
g  f (0)
=
=
f  f ( x  1) =
 f  ( g  f )( x)
=
Example
Let f : R  R be a function defined by f ( x)  x 2  1 .
Find
f [2,1) , f [0,1], f 1 [0,5] , f 1 [10,26] .
Example
Let f ( x ) 
2x  7
, evaluate f  f    f (x ) .

x 1
1997 times
1.4
Constant Function and Identity Function
Definition
Example
Definition
Let A and B be two sets and b  B . A function (or mapping) f : A  B is called a
constant function if and only if f (a )  b, a  A .
f : R  R defined by f ( x )  2, x  R is a constant function.
A function f : A  A is called an identity function of A if and only if f (a )  a, a  A .
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Functions
Definition
Advanced Level Pure Mathematics
Let A, B be two sets. f and g are two real-valued functions defined on A and B
respectively. Then,
Example
(1) ( f  g )( x )  f ( x )  g ( x ),
x  A  B;
(2) ( f  g )( x )  f ( x )  g ( x ),
x  A  B;
(3) (cf )( x )  cf ( x ),
x  A;
(4) ( fg )( x )  f ( x ) g ( x ),
x  A  B;
 f
f ( x)
,
(5)  ( x ) 
g( x)
g
x  A  B \ x : g( x)  0
(1) Let f ( x ) 
x , x  0 and g ( x ) 
Then ( f  g )( x )  f ( x )  g ( x ) 
x 2  1,  x  1.
x  x 2  1,  x  1.
(2) Let f ( x )  log x, x  0 and g ( x )  sin x, x  R.
f
log x
, x  (0,) \ n : n  N .
Then  ( x ) 
sin x
g
1.5
Injective, Surjective and Bijective Functions
Definition
Let f : A  B be a function (or mapping) . f is called an injection ( injective function,
one to one function ) if and only if the following holds:
a1 , a2  A, if f (a1 )  f (a2 ), then a1  a2 .
( or equivalently, a1 , a2  A, if a1  a2 , then f (a1 )  f (a2 ) ).
Example
(1) The function f ( x )  log 10 x,
( x  0) is injective from the set (0,) to R .
(2) The function g( x)  x 2 ,
x  R is not injective from R to R ,
since it is easy to find x1  x 2 but g( x1 )  g( x 2 ) .
Now, if the domain of g is restricted to [0,) ,
it would become injective.
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Functions
Example
Advanced Level Pure Mathematics
Let f : R  R be a function defined by
f ( x )  sin x
(x  R)
Is f injective? Why?
Solution
Example
Let f : R  R be a function defined by
f ( x)  x  3
(x  R)
Prove that f is injective.
Solution
Example
Let f : R 2  C be a function defined by
f ((a , b))  a  bi
Prove that f is injective.
((a, b)  R 2 )
Solution
Example
Let a , b, c, d be real numbers and c  0 .
ax  b
 d
f : R \    R be a function defined by f ( x ) 
.
cx  d
 c
Show that if ad  bc  0 , then f is injective.
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Functions
Definition
Advanced Level Pure Mathematics
Let f : A  B be a function (or mapping) . f is called a surjection (surjective function,
onto function) if and only if the following holds:
Pr2 f  B
(or equivalently, b  B, if a  A such that b  f (a ). )
Example
(1) The function f ( x )  x 3 , x  R is surjective from R to R .
(2) The function g ( x )  sin x, x  R is not surjective from R to R , but is surjective
from R to [1,1] .
Example
Let f : R  R be a function defined by
f ( x)  x 2
Prove that f is not surjective.
Solution
(x  R)
For any x  R, x 2  0 .
Hence, the pre-image of any negative element of the range R does not exist.
For example, 1  R but the pre-image of  1 under f is purely imaginary.
That is the pre-image of  1 under f does not belong to R , and so f is not surjective.
Remark
If the range of f is change to
Then f becomes surjective.
Example
Show that the function f : R  (0,1] defined by f ( x ) 
1
is surjective.
x 1
2
Solution
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Functions
Example
Advanced Level Pure Mathematics
Let f : C  C be a function satisfying f (az1  bz 2 )  af ( z1 )  bf ( z 2 ) for any real
numbers a and b and any z1 , z 2  C . Show that
(a) f (0)  0 ,
(b) f is injective if and only if when f ( z )  0 we have z  0 .
Solution
Definition
Let f : A  B be a function (or mapping) . f is called a bijection ( bijective function or
one-one correspondence) if and only if f is both injective and surjective.
Let f : A  B be a function, the set S  ( x, y) : ( y, x)  f  may not be a function from B and A .
In order to make S also a function, f itself must be bijective.
The function so formed is known as inverse function.
Definition
Let f : A  B be a bijective function (or a bijective mapping), the set
( x, y) : ( y, x)  f 
is defined as the inverse function or inverse mapping of the function f , denoted by f 1 ,
i.e. f 1  ( x, y) : ( y, x )  f  and f 1 is then a function from B to A .
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Functions
Advanced Level Pure Mathematics
Remark
The inverse function of a bijective function is also bijective function.
Example
(1) Let A  1,2,3 , B  x, y, z, and f  (1, x), (2, y), (3, z) is a bijective function from
A to B . The inverse of f is f 1  ( x,1), ( y,2), ( z,3).
(2) The function f ( x )  2 x  1 , x  R is bijective from R to R . Then the inverse
function of f is f 1 ( x ) 
x 1
, x  R .
2
Find the inverse function.
(3) The function f ( x )  log 10 x , x  0 is a bijection from R  to R .
Then the inverse function of f is f 1 ( x)  10 x , x  0 .
(4) The function f ( x )  x 2 , x  [0,) is a bijection from [0,) to [0,) .
Then the inverse function of f is f
Example
1
( x) 
x  0.
Each of the following is a function from R to R . State whichj one is injective (one to one),
Which is surjective (onto) and which is bijective ( one-one correspondence)?
(a)
(d)
f ( x)  10
x
f ( x )  x ( x 2  1)
(b)
(e)
f ( x)  x  1
f ( x )  x 2 sin x
(c)
(f)
 x2 1
,

x

1
f ( x)  
 1,

x 1
x 1
f ( x)  x 3  1
Solution
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Functions
1.6
A.
Advanced Level Pure Mathematics
Some Special Real Functions
Even and Odd Functions
Definition
Example
Example
A function f ( x ) is said to be an even function if f ( x )  f ( x ) .
f ( x)  x , f ( x)  x 2 , f ( x)  cos x on R are even functions.
Prove that the function f ( x )  x sin x on R is an even function.
Solution
Remark
Graph of an Even Function
Definition
A function f ( x ) is said to be an odd function if f ( x )   f ( x ) .
Example
Remark
f ( x )  x 3 , f ( x )  sin x on R are odd functions.
Graph of an Odd Function

the graph must be passing through
the origin

the graph remains the same when it
is rotated anti-clockwise ( or
clockwise ) through an angle of
 ( we say that the graph is
symmetrical about the origin)
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Functions
Theorem
Advanced Level Pure Mathematics
Properties of Even and Odd Functions
(1) The sum of two even functions is even.
(2) The sum of two odd functions is odd.
(3) The product of two even functions is even.
(4) The product of two odd functions is even.
(5) The product of an even function and an odd function is odd.
Proof
Example
Given that f : R  R such that x , y  R
f ( xy )  f ( x )  f ( y ) .
Find f (1) and f (1) .
Hence show that f is even.
Solution
Example
Given that f : R  R such that x , y  R
f ( x  y )  f ( x  y )  2 f ( x ) f ( y ) where f (0)  0 .
Show that f is even.
Solution
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Page 13
Functions
Example
Advanced Level Pure Mathematics
Let f be a function on R such that
f ( x  y )  f ( x ) f ( y )  f (a  x ) f (a  y )
( x , y  R)
where a is a positive constant.
If f (0)  1, find f (a ) .
Hence show that f is even.
Solution
B.
Bounded Functions
Let f be a function defined on the set of real numbers A and M be a positive constant.
(1) For any x  A , if
f ( x)  M
then f is said to be bounded from above on A . M is called an upper bound of f .
(2) For any x  A , if
f ( x)  M
then f is said to be bounded from below on A . M is called an lower bound of f .
(3) For any x  A , if
f ( x)  M
then f is said to be bounded on A .
Obviously, a function which is bounded from above and also bounded from below must be a bounded
function.
From the definition of bounded function, it is obvious that the graph of a bounded function lies between two
horizontal straight lines y  M and y   M as shown in figure.
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Functions
Example
Advanced Level Pure Mathematics
The function f ( x )  x 2 on R is bounded from below by 0 since x 2  0 .
However, it is not bounded from above.
Example
The trigonometric functions f ( x )  cos x and f ( x )  sin x on R are bounded functions
since
sin x  1 and cos x  1
f ( x )  tan x, f ( x )  sec x, f ( x )  csc x are not bounded functions.
Solution
Example
Let f : R  R be a real function such that
f ( x  y)  f ( x) f ( y)
(x , y  R)
(a) Show that f is bounded from below by 0 .
(b) Furthermore, if f is not identically equal to zero,
(i) find f (0).
(ii) show that f ( x )  0
(x  R)
Solution
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Functions
C.
Advanced Level Pure Mathematics
Monotonic Functions
Definition
Let f be a function defined on the set of real numbers A .
(1) f is said to be monotonically increasing on A if and only if
a  b  f (a )  f (b )
(a, b  A)
On the other hand, f is said to be strictly increasing on A if and only if
a  b  f (a )  f (b )
(a, b  A)
(2)
f is said to be monotonically decreasing on A if and only if
a  b  f (a )  f (b )
(a, b  A)
On the other hand, f is said to be strictly decreasing on A if and only if
a  b  f (a )  f (b )
(a, b  A)
Example
By sketching the graphs of y  x and y  x 2 , it is obvious that y  x is an increasing
function on R while the function y  x 2 is decreasing on (,0] and increasing on
[0,) .
Solution
Example
Show that the function y 
1
is strictly decreasing on (0,) .
x
Solution
Theorem
Let f : R  R be a bijective function.
If f is strictly increasing, then f 1 is also strictly increasing;
If f is strictly decreasing, then f 1 is also strictly decreasing.
Proof
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Functions
Example
Advanced Level Pure Mathematics
Let f : R  R be bijective and a1  a 2    a n , where n  2 .
(a) Suppose f is strictly increasing. Prove that its inverse f 1 is also strictly incresing
1 n

a1  f 1   f (a k )   a n .
 n k 1

(b) Define h( x )  pf ( x )  q, where p, q  R and p  0 .
and deduce that
Show that h 1 ( x )  f 1 (
xq
)
p
1 n

1 n

and deduce that h 1   h(a k )   f 1   f (a k ) 
 n k 1

 n k 1

Solution
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Functions
D.
Advanced Level Pure Mathematics
Periodic Functions
Definition
Let y  f ( x ) be a function defined on R . If there exists a positive constant T such that
f ( x  T )  f ( x)
(x  R)
f ( x ) is called a periodic function with period T .
From definition,
f (x  T) 
Similarly,
f ( x  2T ) 
f ( x  nT )  f ( x ) ,
where n is an integer,
and so if T is the period of a periodic function f ( x ) , then any multiple of T is also a period of f ( x ) .
Hence, we have
In general, we select the smallest positive period as the period of the periodic function.
Example (a) If f ( x ) is a periodic function with period T , prove that f (wx ) is also a periodic function
T
.
w
(b) Find the period of the function y  sin( wx   )
of period
Solution
Theorem
Properties on Combining Periodic Functions
Let y  f ( x ) and y  g ( x ) be two periodic functions with the smallest positive periods
S and T respectively.
S
f
If
is a rational number, then f  g, f  g, fg and
are periodic functions.
T
g
Proof
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Page 18
Functions
Example
Advanced Level Pure Mathematics
(a) Suppose f : R  R is a function satisfying f (a  x )  f (a  x ) and
f (b  x )  f (b  x ) for all x , where a, b are constants and a  b .
Let w  2(a  b) . Show that w is a period of f , i.e., f ( x  w )  f ( x ) for all x  R .
(b) Suppose g : R  R is a periodic function with period T  0 satisfying g ( x )  g ( x )
for all x . Show that there exists c with 0  c  T such that g (c  x )  g (c  x ) x .
Solution
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Page 19
Functions
Example
Advanced Level Pure Mathematics
Given that f : R  R satisfies the following relation
f ( x  a) 
1

2
f ( x )  [ f ( x )] 2 ,
where a is a real constant.
Show that f ( x ) is a periodic function with period 2a .
Solution
Example
Let f : [1,1]  [0,  ] , f ( x )  arc cos x and g : R  R , g ( x )  f (cos x ) .
Show that g ( x ) is even and periodic.
Solution
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Page 20
Functions
1.7
I.
Advanced Level Pure Mathematics
Elementary Functions
Constant Function
A constant function is a function of which the image of every
element of the domain is the same. That is, if c is a constant,
then y  c is a constant function.
II. Exponential Function
Let a be a real constant. Then the function y  a x (0  a  1) is
an exponential function.
(1) The domain of an exponential function is R .
(2) The graph of an exponential function is shown in the figure:
(3) When 0  a  1 , the exponential function a x is strictly
decreasing.
When a  1 , the exponential function a x is strictly
increasing.
(4) In particular, when a  e , an irrational number which is approximately equal to 2.7182828, we write
y  exp x to denote the function y  e x .
III. Logarithmic Function
Let a be a real constant. Then the function y  log a x , (0  a  1)
is called a logarithmic function.
In particular, when a  e , the logarithmic function log e x or
sometimes denoted by ln x , is called the natural logarithm of x . It is
the inverse function of y  exp x and is another important function
in mathematical analysis.
IV. Greatest Integer Function
The greatest integer function, denoted by y  [ x ] , is the
greatest integer less than or equal to x .
That is, y  [ x ]  n (n  x  n  1, n  Z )
For example, [3.1]  3, [ 4]  4 , [0.5]  1, [3.5]  4
Its graph is shown in the figure:
Prepared by K. F. Ngai
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Functions
1.8
Example
Advanced Level Pure Mathematics
Revision Exercise
Let f : R  R be a real function such that x, y  R, f ( x  y )  f ( x )  f ( y ).
(a) Show that f (nx )  nf ( x ) for all positive integers n  1 .
(b) If f is not identically equal to zero, show that f is not bounded.
Solution
Prepared by K. F. Ngai
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Functions
Example
Advanced Level Pure Mathematics
A function f : R  R is said to be additive if f ( x  y )  f ( x )  f ( y ), x, y  R .
(a) Let f be an additive function.
(i)
Show that for all integers n , f (nx )  nf ( x ).
Hence deduce that f (rx )  rf ( x ) for any rational number r .
(ii) By using the first result of (a)(i) show that if f is also bounded on R , then
f ( x ) is identically equal to zero.
(b) Suppose g is an additive function and is bounded on the interval [0, a ] , where a is a
positive real constant. Let h : R  R be a function defined by h( x ) 
g (a )
x  g( x) ,
a
x  R.
(i)
Show that h is additive and bounded on [0, a ] .
(ii)
Show that h is a periodic function with period a .
Hence deduce that h is a bounded function.
(iii)
Prove that g ( x ) 
g (a )
x.
a
Solution
Prepared by K. F. Ngai
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