Let S be the domain of the function f(x) = _ 1/2 ( x^2 - 4x + 3 x^2 - 9 ) + ⁻¹ ( 2x+1 x-2 ) . The number of elements in the set x Z : 0 x 10 and x S is :

Step-by-step solution

For the first term to be defined, the argument of the square root must be non-negative, and the argument of the logarithm must be strictly positive: _ 1/2 ( x^2 - 4x + 3 x^2 - 9 ) 0 Since the base is 1 2 0 Factorising the rational expression: 0 This requires x 3 . Cancelling (x-3) , we get: 0 From x-1 x+3 1 , we have x-1 - (x+3) x+3 0 -4 x+3 0 x+3 > 0 x > -3 . From x-1 x+3 > 0 , since x+3 > 0 , we must have x-1 > 0 x > 1 . Thus, the condition for the first term is x (1, ) 3 . For the second term to be defined, the