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Composition of Functions

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Composition of Functions - Lesson Summary

Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by gof, is defined as function gof : A → C given by

gof (x) = g(f (x)), ∀ x ∈ A.

Points to remember:

(i) If f:A → B and g:B → C, then gof:A → C.
(ii) Function gof is possible only if the range of function f is the domain of function g.
(iii) When gof is possible, fog may or may not be possible.
(iv) If both gof and fog are possible, then they may or may not be equal.

Ex:
Suppose you are at a furniture shop and the shopkeeper giving a discount of 10 percent on the purchase of furniture exceeding 5000 rupees.

Sol:
Let us now to calculate the discount if a purchase of x rupees is made.
Amount over Rs 5000= x – 5000

Let g: R → R such that g(x) = x – 5000
Discount = 10 % of (x – 5000)

Let f: R → R such that f(x) = 10% of (x – 5000)
Let x = 8000
fog(8000) = f(g(8000)) ('.' g(x) = x – 5000)
= f(8000–5000)
=f(3000) = 10 % of 3000 = 300 ( f(x) = 10% of (x – 5000) ).

Properties of Composite Functions
Let f: R → R and g: R → R defined by f(x) = 2x and g(x) = 3x2.

To find: gof and fog

gof(x) = g[f(x)]

= g[2x] ['.' f(x) = 2x]

= 3(2x)2 ['.' g(x) = 3x2]

gof(x) = 12x2

fog(x) = f[g(x)]

= f[3x2] [Since, g(x) = 3x2]

= 2(3x2) [ '.'  f(x) = 2x]

fog(x) = 6x2

Observe that gof(x) ≠ fog(x)

Hence, we can say that the two functions gof and fog are unequal.

The composition of functions does not satisfy the commutative property.

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