unit 1 differential calculus - IGNOU

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(ii) = , if x ≠ 0
x | x|
(iii) | x − y | ≤ | x | + | y |
(b) State whether the following are true or false.
(i) 0 ∈ [1, ∞] True/False
(ii) − 1 ∈ (− ∞, 2) True/False
(iii) 1 ∈ [1, 2] True/False
(iv) 5 ∈ (5, ∞) True/False
1.3 FUNCTIONS
Now let us move over to functions. Here we shall present some basic facts about
functions which will help you refresh your knowledge. We shall look at various
examples of functions and shall also define inverse functions. Let us start with the
definition.
1.3.1 Definition and Examples
Definition 3
If X and Y are two non-empty sets, a function f from X to Y is a rule or a
correspondence which connects every member of X to a unique member
of Y. We write f : X → Y (reads as “f is a function from X to Y” or “f is a
function of X into Y”). X is called the domain and Y is called the co-domain
of f. We shall denote by f (x) that unique element of Y, which corresponds
to x ∈ X.
The following examples will help you in understanding this definition better.
Example 1.1
Consider f : N → R defined by f (x) = − x. Is “f ” a function?
Solution
“f ” is a function since the rule f (x) = − x associates a unique member (− x)
of R to every member x of N. The domain here is N and the co-domain is R.
Example 1.2
x
Consider f : N → Z, defined by the rule f ( x)
= . Is “f ” a function?
2
Solution
“f ” is not a function from N to Z as odd natural numbers like 1, 3, 5 . . .
from N cannot be associated with any member of Z.
Example 1.3
Differential Calculus
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