Consider the functions f(x) = x2 - 6x and g(x) = x - 6
If f(x) and g(x) are the two functions then the composite function is denoted as (f o g)(x) and it is defined as,
(f o g) (x) = f(g(x)
(1)
(f o g) (x)
By using the definition of composite function,
(f o g) (x) = f(g(x))
Here f(x) = x2 - 6x and g(x)= x -6
f(g(x)) = f (x - 6)
Replace x by x - 6 in f(x)
f(g(x)) = (x - 6)2 - 6(x - 6)
Expanding this,
f(g(x)) = x2 - 12x + 36 - 6x + 36
Therefore, (f o g) (x) = x2 - 18x + 72
(2)
(g o f) (x)
By using the definition of composite function,
(g o f) (x) = g(f(x))
Here f(x) = x2 - 6x and g(x)= x - 6
g(f(x)) = g (x2 - 6x)
Replace x by x2 - 6x in g(x)
g(f(x)) = (x2 - 6x) - 6
Expanding this,
g(f(x)) = x2 - 6x - 6
Therefore, (g o f) (x) = x2 - 6x - 6
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