unit 1 differential calculus - IGNOU

(iii) f : R → R defined by f (x) = x 2 + 1.
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(iv) f : X → R defined by f (x) = where X stands for the set of
x
non-zero real numbers.
x + 1
(d) Show that the function f : X → X such that f ( x)
= where X is
x − 1
the set of all real number except 1, is one-one and onto. Find its
inverse.
(e) Give one example of each of the following :
(i) a one-one function which is not onto?
(ii) onto function which is not one-one?
(iii) a function which is neither one-one nor onto?
1.3.4 New Functions from Old
In this sub-section, we shall see how we can construct new functions from some
given functions. This can be done by operating upon the given functions in a
variety of ways. We give a few such ways here.
1.3.5 Operations on Functions
Scalar Multiple of a Function
Consider the function f : x → 3x 2 + 1, for all x ∈ R. The function
g : x → 2 (3x 2 + 1) for all x ∈ R is such that g (x) = 2f (x), for all x ∈ R. We
say that g = 2f and that g is a scalar multiple of f by 2. In the above
example, there is nothing special about the number 2. We could have taken
any real number to construct a new function from f. Also, there is nothing
special about the particular function that we have considered. We could as
well as have taken any other function. This suggests the following
definition : Let f be a function with domain D and let k any real number.
The scalar multiple of f by k is a function with domain D. It is denoted by kf
and is defined by setting (kf) (x) = kf (x).
Differential Calculus
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