For the following two (02) items: Consider the function f(x) = x 1 - x ( x > 0 , x 1 ). What is (1 - x)f( x ) + xf( x + 1) equal to?

Step-by-step solution

Given f(x) = x 1 - x Substitute x in f(x) : f( x ) = x 1 - x Substitute x + 1 in f(x) : f( x + 1) = x + 1 1 - ( x + 1) = x + 1 - x Now, evaluate the given expression: (1 - x)f( x ) + xf( x + 1) = (1 - x) ( x 1 - x ) + x ( x + 1 - x ) Using 1 - x = (1 - x )(1 + x ) : = (1 - x )(1 + x ) ( x 1 - x ) - x ( x + 1) = x (1 + x ) - x ( x + 1) = 0 Answer: 0