Page 14 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Let us now discuss another important concept in this section. This is about the composition or
combination of two function. Consider the following situation:
Let S = {1, 2, 3, 4}, T = {1, 4, 9, 16}, N = {1, 2, 3, 4....} be any three sets Let f: ST be defined by f(x) = x ,
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 x  S and g: TN be defined by g(x) = x +1,  x  T. Then, by the function f, an element x  S
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is mapped to f(x) = x . Further by the function g the element f(x) is mapped to f(x) + 1 = x + 1.
Denote this as g(f(x)). Define a function h: SN by h(x) = g(f(x)). This function h maps each x in
S to some unique elements g(f(x)) = x + 1 of N. The function h is called the composition or the
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composite of the functions f and g. Thus, we have the following definition:
Definition 10: Composite of Functions
Let f: ST and g: TV be any two functions. A function h: SV denoted as h = gof and defined by
h(x) = (gof) (x) = g(f(x)),  x  S
is called the composite of f and g.
Note that the domain of the composite function is the set S and its co-domain is the set V. The set
T which contains the range of f is equal to the domain of g.
But in general, the composition of the two functions is meaningful whenever the range of the
first is contained the domain of the second.

Example: Let S = T = {1, 2, 3, 4...}, Define
f(x) = 2x and g(x) = x + 5. Then

“gof is defined as (gof) (x) = g(f(x)) = g(2x) = 2x + 5.
Note that we can also define fog the composite of g and f. Here (fog) (x) = f(g(x)) = f(x + 5) =
2 (x + 5) = 2x = 10, Also (fog) (1) = 12 and (gof) (1) = 7. This shows that ‘fog’ need not be equal to ‘gof’.
Let S = {1, 2, 3} and T = {a, b, c). Let f: ST be f(1) = a, f(2) = b, f(3) = c. Define a function g: TS as
g(a) = 1, g(b) = 2 and g(c) = 3. Under the function g, the element f(x) in T is taken back to the
element x in S. This mapping g is called the inverse of f and is given by g(f(x)) = x, for each in S.
You may note that f(g(a)) = a, f(g(b)) = b and f(g(c)) = c. Thus, we have the following definition:
Definition 11: Inverse of a Function
Let S and T be two non-empty sets. A function f: ST is said to be invertible if there exists a
function g: TS such that

(gof)(x) = x for each x in S,
and
(fog)(x) = x for each x in T.
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In this case, g is said to be the inverse of f and we write it as g = f .

Did u know? Do all function possess inverses?
No, all functions do not possess inverses. For example, let S = {1, 2, 3} and T = {a, b). If f: ST
is defined as f(1) = f(2) = a and f(3) = b, then f is not invertible. For, if g: ST is inverse of f,
then

(gof) (1) = g(f(l)) = g(a)
and (gof) (2) = g(f(2)) = g(a).
Therefore, 1 = 2, which is absurd.

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