Syllabus
· Examples
of limit of functions
· Properties
of limits of functions
· One
sided limits
· Infinite
limits and limit at infinity
· Theorems
of continuity of functions
· Different
types of discontinuity of functions
· Combination
of continuous functions and composite functions
· Bolzano's
theorem
· Uniform
Continuity
· Lipschitz
condition
· Monotone functions
Q.1.
Define the following terms:
a. Function
b. Domain, range and co–domain of a
function
c. Image and pre–image of a function
d. Some special types of function
e. Composite function
f. Inverse function
g. Bounded and unbounded function on a set
h. Monotonic real valued function on a set
Soln: a. Function: Let X and Y be two non–empty sets. If there exits a rule f which associates to every element a Î X, to a unique element b ÎY, then such a rule f is called a function from the set X to the set Y.
The function from set X to set Y in
symbol can be written as
f: X ®
Y. Example: Let X = {1, 2, 3} and Y = {5, 7, 9, 11}. Let f from X to Y be
defined by f (1) = 5, f(2) = 7, f(3) = 9, then f is a function and f = {(1, 5),
(2, 7), (3, 9)}. This function can also be written as f(x) = 2x + 3, "
x Î
X.
b. Domain, range and co-domain of a function: If f: X ® Y, then the set X is called the domain of function f and the set of all y Î Y which corresponds to some x Î X is called the range of the function f, and is denoted by f(X),
i.e. f(X) = {f(X): x Î X} Í Y.
If f: X ® Y, then the set of all y Î Y is called the co–domain of f. Clearly, f(X) Í Y, i.e. the range is subset of co–domain.
c. Image and pre-image of a function: If f: X ®Y be a function. For every xÎX, there exists a unique y Î Y such that f(x) = y. Then y = f(x) is said to be the image of x and x is said to be pre-image of y under the function f.
d. Some
special types of function:
i) One to one or injective function: A function f from a set X to another set Y, i.e. f: X ®Y is said to be one to one or injective if distinct elements (or pre-images) in X have distinct images in Y.
In symbol, f : x ® y is said to be one to one if for any
x1, x2 Î X,
x1 ¹ x2 Þ f(x1) ¹ f(x2)
or, equivalently,
f(x1) = f(x2) Þ x1 = x2
In other words, a function f is said to be one to one or injective if (x1, f(x1)), (x2, f(x2))Î f Þ x1 = x2.
Thus, under one to one function all elements of X are related to different elements of Y.
ii) Onto or surjective function: A function f from a set X to another set Y, i.e. f: X ® Y is said to be onto or surjective, if every element of Y, is an image of at least one element of X, i.e. every element of Y has a pre-image, or f(X) = Y.
iii) One to one and onto or bijective function: A function that is both one to one and onto (i.e. injective and surjective) is called bijective function.
Consider a function f: X ® X defined by f(x) = x " x Î X. Obviously, it is well-defined and both one to one and onto, i.e. it is bijective function. This function is known as the identity function.
iv) Constant function: If f : X ® {a}
be defined by
f(x) = a "
x Î
X, then f is said to be a constant function on X. That is f(X) = {a} in this
case.
e. Composite function: If f : X ® Y and g : Y ® Z, then the composite of functions f and g, written g o f, is the function which maps each element xÎX into (gof)(x) = g(f(x)) Î Z.
The functions obtained by means of composition of functions, as above are called composite functions.
Example: Let f: R®R+
be defined f(x) = x2 + 1 " x Î R, and
g Î
R ®
R be defined by g(x) = 2x + 1 " xÎR. Then the function g o f: R ® R+ is defined
by
(g o f) (x) = g(f(x))
= g(x2 + 1)
=2(x2 +1) + 1
= 2 x2 + 3, " x Î R.
Also, function f o g: R ® R+ exists and is defined by
(f o g) (x) = f(g(x))
= f(2x + 1)
= (2x + 1)2 + 1
= 4x2 + 4x + 2, " x Î R.
f. Inverse
function: Let f: X ® Y be one to one and onto function. Then the function g
:Y ®
X which associates to each element
b Î
Y to the unique element a Î X such that f(a) = b is called the inverse function of
f. The inverse function of f is denoted by f –1.
It should be noted that every
function does not have an inverse.
A function f : X ®
Y has inverse iff f is one to one and onto. If function f has inverse, then f
is said to be invertible and written as f –1 : Y ® X.
Also, if aÎX,
then f(a) = b where b Î Y
Þ a = f –1(b).
Example; Let f : R® R be defined by f(x) = 2x – 1.
To find f –1, first we have to prove that f is one to one and onto.
Let x1, x2Î R such that x1 ¹ x2.
Now, x1 ¹ x2 Þ 2x1
¹
2x2
Þ 2x1 –1 ¹ 2x2 –1
Þ f(x1) ¹ f(x2).
Therefore, f is one to one.
g. Bounded and unbounded function on a set: Let f be a function with non-empty domain set S and range set in R. Then f is said to be bounded on S, below or above according as there exist real numbers k1, k2 such that f(x1) ³ k1 or £ k2, " x Î S. The numbers k1 and k2 are known as the lower and upper bounds of f on S.
If a function f is bounded both below and above on a domain set S, then it is said to be a bounded function on S; otherwise if it is neither bounded below, nor above then it is said to be unbounded on S.
For example f(x) = x2 is bounded on S = [2, 4] for 4 £ f(x) £ 16 " x Î [2, 4].
The function f(x) = 1/x is
unbounded on S = (0, 1),
for 1 < f(x) < ¥ " x Î (0, 1).
h. Monotonic
real valued function on a set: If S
Ì
R and f : S ®
R, then f is said to be monotonically non–decreasing or non-increasing on S
according as f(x1) ³ or £ f(x2) " x1, x2
Î
S such that x1 > x2.
A function which is either monotonically non–decreasing or non-increasing on a subset S of R is called a monotonic function on S. Particularly, f is said to be monotonically increasing or decreasing on S according as f(x1) > or < f(x2) " x1, x2 Î S such that x1 > x2.. A function which is either monotonically increasing or decreasing on set S is called strictly monotonically on S.
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