If f(x) is a linear function where f(x+2)- f(x) = 6 , find the value of : f^-1(x+2) - f^-1(x).
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f ( x ) is linear function,
let f( x ) =a x + b
f(x +2 ) - f ( x ) = 6
a(x + 2) + b -ax - b = 6
a(2) = 6
a = 3
hence,
f(x ) = 3x +b
f(x ) -b/3 =x
so, f-1( x) = (x -b)/3
now,
f -1 ( x +2 ) - f-1(x )
=(x + 2 -b)/3 -(x - b)/3
=2/3
let f( x ) =a x + b
f(x +2 ) - f ( x ) = 6
a(x + 2) + b -ax - b = 6
a(2) = 6
a = 3
hence,
f(x ) = 3x +b
f(x ) -b/3 =x
so, f-1( x) = (x -b)/3
now,
f -1 ( x +2 ) - f-1(x )
=(x + 2 -b)/3 -(x - b)/3
=2/3
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