A function is a relation that assigns to each input value exactly one output value. A function can be defined as a rule that relates how one quantity depends on other quantities.
Composition of functions
Composition of functions refers to application of output from one function as input for another function. This means the range (y-values or output) of one function becomes the domain (x-values or input) of the other function. It is written as:
(f ͦ g)(x) = f(g(x))
and read as ‘f of g of x’ or ‘f composed with g of x’.
Example:
f(x) = 3x – 4 and g(x) = x2. Find (f ͦ g)(x) and (g ͦ f)(x).
(f ͦ g)(x) = f(g(x))
f(x2) = 3x2 – 4
(g ͦ f)(x) = g(f(x))
g(3x – 4) = (3x – 4)2
Example:
f(x) = {(-2, 2), (-1, 1), (0, 0), (1, -1)} and g(x) = {(-2, -3), (-1, -2), (1, 0), (2, 1)}. Find (f ͦ g)(-1).
(f ͦ g)(-1) = f(g(-1))
From ordered pair (-1, -2) of g(x)
g(-1) = -2
Now, f(g(-1)) = f(-2)
So, from ordered pair (-2, 2) of f(x)
f(-2) = 2
Hence, (f ͦ g)(-1) = f(g(-1) = f(-2) = 2
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