1.2. FUNCTIONS
1.2
1.2.1
11
Functions
What is a Function?
In this section, we only consider functions of one variable. Loosely speaking, a
function is a special relation which exists between two variables. In introductory
mathematics classes, the de…nition below is the one which is usually given for a
function.
De…nition 22 (function) Let A and B denote two sets.
1. A function f from A to B is a rule which assigns a unique y 2 B to each
x 2 A. We write y = f (x). f (x) denotes the value of the function at x.
2. x is called the independent variable (also called an input value), y is the
dependent variable (also called an output value). Remember, if we say
that y is a function of x, it implies that it depends on x.
3. The domain of f is A, the set of values of x. It is also denoted D (f ) or
Dom f . When the domain of a function is not given, it is understood to
be the largest set of real numbers for which the function is de…ned.
4. The range of f is the set R (f ) = ff (x) : x 2 D (f )g. It is also denoted
Range f .
A function can also be de…ned in terms of pairs. In this manner, we simply
give the pairs of elements which are in relation. More precisely,
De…nition 23 (function) Let A and B denote two sets. A function from A
to B is a subset of A B that is a set of ordered pairs with the property that
whenever (a; b) 2 f and (a; c) 2 f then b = c.
When we say that (a; b) 2 f , it means that b = f (a). With this form of the
de…nition, the domain of f is simply the set of …rst members of each pair, the
range is the set of second members. A set of ordered pairs is a function if no
two pairs have the same …rst member and di¤erent second member.
At this stage, it is also important for students to understand the di¤erence
between f and f (x). f is a function. As we have seen in the above de…nition,
it is a set of ordered pairs. f (x) is an element of R (f ), the range of f .
A function from A into R is called real-valued.
De…nition 24 (mapping) If f is a function from A to B, we also say that f
is a mapping from A into B, or that f maps A into B. We often write:
f :A!B
De…nition 25 (image) If f is a function from A to B and y = f (x) (or if
(x; y) 2 f ), then we say that y is the image of x under f .
The following proposition follows from the de…nition of a function.
12CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU
Proposition 26 (vertical line test) A graph is the graph of a function if it
passes the vertical line test, that is if no vertical line can intersect the graph in
more than one point.
Example 27 Let A = f0; 1; 2; 3; 4; 5g and B = R. Let
f = f(0; 6) ; (1; 54) ; (2; 70) ; (3; 54) ; (4; 6) ; (5; 74)g
It is easy to see that f is a function. Each x 2 A belongs to exactly one pair in
f . The fact that some pairs have the same second component does not contradict
the de…nition of a function. Though it indicates something else we will study
soon.
Example 28 With A and B as in the previous example,
g = f(0; 6) ; (1; 54) ; (1; 74) ; (2; 70) ; (3; 54) ; (4; 6) ; (5; 74)g
is not a function. The pairs (1; 54) and (1; 74) have the same …rst component
but have di¤ erent second components. This violates the de…nition of a function.
Example 29 Often, a function is given by a formula which gives the relationship between input and output values as in this example. Let A = B = R. We
de…ne the function h by
h = (x; y) 2 R2 : y = x2 + 1
In other words, h (x) = x2 + 1.
Example 30 The identity function, denoted i, is de…ned on any non-empty set
A by:
i = f(x; x) : x 2 Ag
In other words, this function maps every element of A into itself, that is i (x) = x
for any x 2 A.
Remark 31 A formula is not enough to de…ne a function. Its domain must
also be speci…ed. Two functions are equal if they have the same set of ordered
pairs. When functions are given by a formula, two functions are equal if they
have the same domain and they de…ne the same relation between input and
output values.
1.2.2
Restrictions and Extensions of Functions
Consider a function f from A to B. Let D1 A. We can de…ne a new function
f1 with domain D1 by :
f1 (x) = f (x) 8x 2 D1
Remark 32 The quanti…er 8 means "for every".
1.2. FUNCTIONS
13
De…nition 33 (restriction) The function f1 as de…ned above is called a restriction of f to D1 .
It is called a restriction because it is de…ned on a smaller set. On the smaller
sets, the two functions agree. In other words, f1 is the same as f , restricted to
D1 .
Example 34 The function
g:
h
;
2 2
x 7 ! sin x
i
! [ 1; 1]
is a restriction of the function
f : R ! [ 1; 1]
to the set
h
;
2 2
i
x 7 ! sin x
.
Remark 35 We restrict functions to a smaller domain when we wish the function to have certain properties it does not have on the larger domain. In the
example above, you will recall from calculus that though sin x is de…ned for all
real numbers, it is not invertible becauseh it doesinot pass the horizontal line test.
However, if we restrict its domain to
; , the function is invertible and
2 2
we haven’t lost any information because on that interval, sin x takes on all its
values. You may recall that sin x is periodic
(see
h
i de…nition below if you have
forgotten what it means). The interval
;
corresponds to one period.
2 2
Instead of restricting the de…nition of a function to a smaller set, we can also
do the opposite. Sometimes, it is useful to extend the de…nition of a function to
a larger set. Let g be a function from A into B, and let D2 be a set containing
A. We can de…ne a new function g2 with domain D2 by:
g2 (x)
=
g2 (x)
= some other function if x 2 D2 n A
g (x) 8x 2 A
De…nition 36 The function g2 as de…ned above is called an extension of g to
D2 .
It is called an extension because we extend the de…nition of g to a larger set.
Example 37 The function
h:R!R
(
x7 !
1
sin x
x
if
x=0
if
x 6= 0
14CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU
is an extension of the function
f : Rn f0g ! R
sin x
x7 !
x
to the set of all real numbers. In this case, the reason for doing this is to
sin x
have a function continuous on the set of real numbers.
is continuous on
x
Rnf0g because it is not de…ned at 0. However, you may recall from calculus that
sin x
sin x
= 1. We can extend
to a continuous function on R with the
lim
x!0 x
x
function of this example.
1.2.3
Operations on Functions
Addition, Subtraction, Multiplication and Division
Usually, a function is de…ned by de…ning the action this function has on elements
of its domain. Let f and g be two functions, call D (f ) the domain of f and D (g)
the domain of g. We de…ne addition, subtraction, multiplication and division
of functions as follows:
The sum of two functions f and g, denoted f + g, is de…ned by:
(f + g) (x) = f (x) + g (x)
The domain of f + g is the set fx : x 2 D (f ) and x 2 D (g)g.
The di¤erence of two functions f and g, denoted f
(f
The domain of f
g) (x) = f (x)
g, is de…ned by:
g (x)
g is the set fx : x 2 D (f ) and x 2 D (g)g.
The product of two functions f and g, denoted f g, is de…ned by:
(f g) (x) = f (x) g (x)
The domain of f g is the set fx : x 2 D (f ) and x 2 D (g)g.
The division of two functions f and g, denoted
f
, is de…ned by:
g
f
f (x)
(x) =
g
g (x)
The domain of
f
is the set fx : x 2 D (f ) and x 2 D (g) with g (x) 6= 0g.
g
1.2. FUNCTIONS
15
Example 38 Given f (x) = x + 1 and g (x) = x
1, …nd f + g, f
g, f g,
f
,
g
…nd their domain.
We …nd these functions by de…ning how they act on elements. Since both f and
g are polynomials, D (f ) and D (g) are the set of real numbers.
(f + g) (x) = f (x) + g (x) = x + 1 + x
the set of real numbers.
1 = 2x. The domain of f + g is
(f g) (x) = f (x) g (x) = x + 1
the set of real numbers.
1) = 2. The domain of f
(f g) (x) = f (x) g (x) = (x + 1) (x
set of real numbers.
(x
1) = x2
g is
1. The domain of f g is the
f (x)
x+1
f
f
(x) =
=
. The domain of is the set of real numbers except
g
g (x)
x 1
g
1.
Composition
Once again, let f be a function from A into B and g be a function from B into
C. The composition of f and g, denoted g f , is the function de…ned by:
(g f ) (x) = g (f (x))
We can also de…ne the composition of two functions in terms of ordered
pairs. let f be a function from A into B and g be a function from B into C.
Then,
g f
=
=
f(a; c) 2 A
f(a; c) 2 A
C : 9b 2 B for which (a; b) 2 f and (b; c) 2 gg
C : c = g (f (a))g
Remark 39 The quanti…er 9 means "there exists".
The domain of g f is the set fx 2 D (f ) : f (x) 2 D (g)g that is it is the set
of elements in the domain of f such that f (x) is in the domain of g.
p
Example 40 Let f (x) = x and g (x) = x2 + 1. Find f g, g f and their
domain.
We …nd f
have:
g by …nding how it acts on an element x. By de…nition, we
(f
g) (x) = f (g (x))
= f x2 + 1
p
x2 + 1
Since x2 + 1 is always de…ned, and always positive, the domain of f g is
all real numbers.
16CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU
We …nd g
have:
f by …nding how it acts on an element x. By de…nition, we
(g f ) (x) = g (f (x))
p
=g x
p 2
=
x +1
=x+1
The domain of g f is fx 2 R : x
0g.
Remark 41 You will notice that in general f
g 6= g f
p
Example 42 Find functions f and g such that (f g) (x) = x 1
It is important to notice that when we write (f
p g) (x), the function g is applied
…rst, then the function f . When we write x 1, …rst, we have to evaluate
x 1, then
p we take the square root. This suggests that g (x) = x 1, and
f (x) = x. If we verify, we get
(f
g) (x) = f (g (x))
= f (x
p
x 1
1)
Remark 43 In general, the notation f n is used to denote composition of f
with itself n times. (f f ) (x) = f (f (x)) = f 2 (x). However, with trigonometric functions, raising to a power means multiplication. For example sin2 x =
2
(sin x) = (sin x) (sin x). Do not confuse this with f (n) which means nth order
derivative.
1.2.4
Injections and Surjections
De…nition 44 (surjection) If f is a mapping of A into B such that R (f ) =
B, then we say that the mapping is onto. We also say that f is surjective, or
that f is a surjection.
To put it simply, a mapping from A into B is a surjection if every element
of B is the image of some element of A.
To prove that a function f from A to B is a surjection, one must prove that
for any y 2 B, there exists an x in A such that y = f (x).
Remark 45 A function f : A ! R (f ) is always a surjection.
Example 46 The function
f : R!R
x 7! x2
1.2. FUNCTIONS
17
is not surjective. However, the function
f : R! [0; 1)
x 7! x2
is surjective.
De…nition 47 (injection) Let f be a function from A into B. f is said to
be injective, or an injection, or one-to-one if one of the three equivalent
conditions below is satis…ed.
1. f (a) = f (b) ) a = b
2. a 6= b ) f (a) 6= f (b)
3. (a; c) 2 f and (b; c) 2 f ) a = b
To put it simply, a mapping from A into B is an injection if di¤erent inputs
produce di¤erent outputs.
Example 48 The function
f : R ! [ 1; 1]
x 7 ! sin x
is
i injection, f (0) = f (2 ) for example. However, its restriction to
h not an
;
is an injection.
2 2
Proposition 49 (horizontal line test) The graph of a function is the graph
of an injective function if it passes the horizontal line test, that is if no horizontal
line can intersect the graph in more than one point.
De…nition 50 A function which is both an injection (one-to-one) and a surjection (onto) is called a bijection.
1.2.5
Inverse Functions
Proposition 51 Let f be a function from A onto B. If f is an injection, then
the function f 1 : B ! A such that f 1 = f(b; a) 2 B A : (a; b) 2 f g is also
a function which is one-to-one.
Proof. The proof of this fact is left as an exercise.
The function f
1
de…ned in terms of f has a name:
De…nition 52 (inverse) Let f be an injective function from A onto B. The
function f 1 = f(b; a) 2 B A : (a; b) 2 f g is called the inverse of f . It is
denoted f 1 .
18CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU
f and f
below.
1
are related in many di¤erent ways. We list a few of these relations
1. D (f ) = R f
1
, R (f ) = D f
2. y = f (x) () x = f
1
3. f
4. f
f
1
1
. This can be seen from the de…nition.
(y). This can be seen from the de…nition.
f (x) = x8x 2 D (f ). See problems at the end of this section.
1
(y) = y8y 2 R (f ). See problems at the end of this section.
You may recall from previous mathematics classes that to …nd the inverse
of a function given by a formula y = f (x), the following steps can be followed:
1. Make sure the function has an inverse (i.e. it is one-to-one).
2. In the relation y = f (x), switch x and y.
3. Solve for y in the relation you obtained above.
4. The new relation you obtained for y is the inverse function.
Example 53 Find the inverse of y = f (x) = 5x + 2.
This is a straight line which is not horizontal, so it passes the horizontal line
test. Hence it is one-to-one. If we switch x and y, we obtain
x = 5y + 2
Next, we solve for y.
x
=
5y + 2 () x
x 2
y=
5
()
2 = 5y
Therefore
f
1.2.6
1
(x) =
x
2
5
Direct Image and Inverse Image of a Set
Let f be a function from A into B. Let E and G be two sets such that E
and G B.
A,
De…nition 54 (direct image of a set) The direct image of E, denoted f (E),
is de…ned by:
f (E) = ff (x) : x 2 Eg
De…nition 55 (inverse image of a set) The inverse image of G, denoted
f 1 (G) is de…ned by:
f
1
(G) = fx 2 A : f (x) 2 Gg
1.2. FUNCTIONS
19
Remark 56 The above de…nition does not require that f be injective or have
an inverse. f 1 (G) is simply the notation for the inverse image of G. The
reader should never think we are talking about the inverse of f .
Remark 57 If should be clear to the reader that f (E)
range of f . Therefore, f (A) = B () f is a surjection.
B. Also f (A) is the
Remark 58 Similarly, it should be clear to the reader that f
1
(G)
A.
Let us consider an example to illustrate this de…nition.
Example 59 Consider f : N ! N de…ned by f (n) = n2
1.
1. Let E = f2; 3; 4g. Find f (E).
By de…nition, f (E) = ff (x) : x 2 Eg in other words,
f (E)
=
=
ff (2) ; f (3) ; f (4)g
f3; 8; 15g
2. Let G = f3; 4; 5; 6; 7; 8g, …nd f 1 (G).
By de…nition, f 1 (G) = fx 2 N : f (x) 2 Gg. The only whole numbers
mapping into an element of G are 2 and 3 since f (2) = 3 and f (3) = 8.
Hence
f 1 (G) = f2; 3g
Certain important set properties are preserved under the direct image or the
inverse image of a set. We list them in two theorems.
Theorem 60 Let f be a function from A into B. Let E and F be subsets of
A. The following is true:
1. If E
F then f (E)
2. f (E \ F )
f (F )
f (E) \ f (F )
3. f (E [ F ) = f (E) [ f (F )
4. f (E n F )
f (E)
Proof. We only prove some of these items. For the remaining ones, see the
problems at the end of this section.
1. Let y 2 f (E). Then, there exists x 2 E such that y = f (x). Because
E F , x is also in F , therefore, y = f (x) is in f (F ).
2. see problems
3. We need to show the inclusion both ways.
20CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU
First, we show that f (E [ F )
f (E) [ f (F ). Let y 2 f (E [ F ).
Then, there exists x 2 E [ F such that y = f (x). Either x 2 E, in
which case y = f (x) 2 f (E). Or, x 2 F , in which case y = f (x) 2
f (F ). Thus, y 2 f (E) [ f (F )
Next, we show that f (E) [ f (F )
f (E [ F ). Since E
E [ F,
by part 1, f (E)
f (E [ F ). Similarly, F
E [ F thus f (F )
f (E [ F ). It follows that f (E) [ f (F ) f (E [ F ).
4. Let y 2 f (E n F ). Then, there exists x 2 E nF such that y = f (x). Then,
x 2 E, thus y = f (x) 2 f (E).
Remark 61 To prove part 4 of the above theorem, we can also use part 1 and
the fact that E n F E.
Theorem 62 Let f be a function with domain in A and range in B. Let G
and H be subsets of B. The following is true:
1. If G
H then f
1
(G)
f
1
(H)
2. f
1
(G \ H) = f
1
(G) \ f
1
(H)
3. f
1
(G [ H) = f
1
(G) [ f
1
(H)
4. f
1
(G n H) = f
1
(G) n f
1
(H)
Proof. We only prove some of these items. For the remaining ones, see the
problems at the end of this section.
1. see problems
2. We need to show the inclusion both ways.
First, we show that f 1 (G \ H) f 1 (G)\f 1 (H). Since G\H
G, by part 1, f 1 (G \ H)
f 1 (G). Similarly, f 1 (G \ H)
1
1
f (H). Thus, f (G \ H) f 1 (G) \ f 1 (H).
Next, we show that f 1 (G) \ f 1 (H)
f 1 (G \ H). Let x 2
1
1
f (G) \ f (H). Then, f (x) 2 G and f (x) 2 H. Therefore,
f (x) 2 G \ H hence, x 2 f 1 (G \ H) by de…nition.
3. see problems
4. see problems
1.2. FUNCTIONS
1.2.7
21
Additional Properties of a Function
General Properties
In this section, we remind the reader of some de…nitions. Theorems about these
properties will be proven later in the chapter.
De…nition 63 Let f : D ! R be a real-valued function. f is said to be:
1. Increasing, if 8a; b 2 D (a
b =) f (a)
f (b))
2. Decreasing, if 8a; b 2 D (a
b =) f (a)
f (b))
3. Strictly increasing, if 8a; b 2 D (a < b =) f (a) < f (b))
4. Strictly decreasing, if 8a; b 2 D (a < b =) f (a) > f (b))
5. Monotone, if it is either increasing or decreasing.
6. Strictly monotone, if it is either strictly increasing or strictly decreasing.
7. Bounded above, if its range is bounded above, that is if 9M 2 R : 8x 2
D f (x) M
8. Bounded below, if its range is bounded below, that is if 9m 2 R : 8x 2
D f (x) m
9. Bounded, if it is both bounded above and below.
10. Even, if 8x 2 D
11. Odd, if 8x 2 D
x 2 D and f ( x) = f (x)
x 2 D and f ( x) =
f (x)
x+T 2D
and f (x + T ) =
x T 2D
f (x). The smallest such T is called the period of the function.
12. Periodic, if 9T 6= 0 2 R : 8x 2 D
13. Lipschitz, if 9k > 0 2 R : 8a; b 2 D
jf (a)
f (b)j
k ja
bj
Global versus Local Properties
Let f : D ! R be a real-valued function. A certain property is said to be a
global property if it is true wherever the function is de…ned. However, certain
properties are only true on an interval of the domain of the function. Such
properties are called local properties. More generally, if f : D ! R is a realvalued function and E
D, we say that f has a certain property on E if the
restriction of f on E has this property.
1
For example, the function f (x) = is not decreasing on its domain which
x
is R n f0g (why?). However, its restriction to (0; 1) is decreasing, so is its
restriction to ( 1; 0).
22CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU
If a 2 D, we say that f has a certain property in a neighborhood of a if
there exists an open set U containing a such that f has the property on U \ D.
For example, the function f (x) = sin x is increasing in a neighborhood of 0;
it is increasing on
1.2.8
2
;
2
.
Exercises
1. Is the subset (x; y) : x 2 R;y 2 R; and x2 + y 2 = 1 a function? Explain.
x2
x
g (x) = x + 1 for x 2 R. Is f = g? Why?
2. De…ne the function f and g by f (x) =
1
for x 2 R and x 6= 1 and
1
3. Prove that if f is an injection from A to B, then f 1 = f(b; a) : (a; b) 2 f g
is a function. Then, prove this function is an injection.
4. Suppose that f is an injection. Show that f 1 f (x) = x for every x
in D (f ). Also, show that f f 1 (y) = y for every y in R (f ).
5. Let f be a function from A into B. Let E and F be subsets of A.
(a) Prove that f (E \ F )
f (E) \ f (F ).
(b) Give an example which shows why the two sets are not equal.
(c) When do you think the two sets are equal, why?
6. Let f be a function with domain in A and range in B. Let G and H be
subsets of B.
H then f
1
(G [ H) = f
1
(a) Prove that if G
(b) Prove that f
1
(c) Prove that f
1
(G
H) = f
f
1
(H)
(G) [ f
1
(H)
(G)
1
(G)
f
1
(H)
7. Give at least one example of a function for each of the 13 de…nition in
de…nition 63. This means you will give at least 13 examples since there
are 13 concepts de…ned. In each case, you will specify the domain of the
function and you will prove that the function satis…es the property you
claim it satis…es.
8. Let A = f 1; 0; 1; 2g and B = N. Which of the following subsets of A
are functions from A into B, explain.
(a) f = f( 1; 2) ; (0; 3) ; (2; 5)g
(b) g = f( 1; 2) ; (0; 7) ; (1; 1) ; (1; 3) ; (2; 7)g
(c) h = f( 1; 2) ; (0; 2) ; (1; 2) ; (2; 1)g
(d) k = f(x; y) : y = 2x + 3; x 2 Ag
B
1.2. FUNCTIONS
23
9. Let f : N ! N be the function de…ned by f (n) = 2n
f 1 (E) for each of the following subsets E of N.
1. Find f (E) and
(a) f1; 2; 3; 4g
(b) f1; 3; 5; 7g
(c) N
10. Let f = (x; y) : x 2 R; y = x3 + 1 .
(a) Let A = fx :
1
x
2g. Find f (A) and f
1
(A).
(b) Show that f is an injection and a surjection.
(c) Find f
1
.
11. For each of the following real-valued functions, …nd the range of the functions f and determine if the function is one-to-one. If f is one-to-one, …nd
the inverse function f 1 and specify the domain of f 1 .
(a) f (x) = 3x
2, D (f ) = R.
(b) f (x) = sin x, D (f ) = fx 2 R : 0
x
g.