Let f:(0, ) [-2, ), f(x)=a x^2-b x+c (where .a, b, c R ) be a surjective function such that _ x 0 f(x)=3 . If g:[1, ) [-2, ), g(x)=f(x) is an invertible function, then identify which of the statement(s) is(are) correct?

Step-by-step solution

aligned f(x) & =5 x^2-10 x+3=5(x-1)^2-2=g(x) f(x) & =a(x-1)^2-2 _ x 0⁺ f(x) & = _ x 0⁺ [a(x-1)^2-2 ]=3 aligned aligned & a-2=3 a=5 & g:[1, ) [-2, ) & g(x)=5(x-1)^2-2 aligned (1) g^ (x)=10(x-1) g^ (1)=0 (2) Domain of g(g(x)) aligned & g(x) 1 5(x-1)^2-2 1 & x 1+ 3 5 & x [1+ 3 5 , ) [1+ p q , ) & q-p=2 aligned (3) aligned & g(x)=g⁻¹(x)=x & 5 (x^2-2 x+1 )-2=x 5 x^2-11 x+3=0 & x= 11 121-60 10 aligned aligned & (4) . d d x [90 (g⁻¹(x) ) ] |_ x=43 = 90 g^ (4) = 90 10(3) =3 & g(x)=43 5(x-1)^2-2=43 x-1=3 x=4 aligned