If f: R R is defined by f(x)= array ccc x+2 x^2+3 x+2 & if & x R- -1,-2 -1 & if & x=-2 0 & if & x=-1 array . then f is continuous on the set

Step-by-step solution

Given that f(x)= array ccc x+2 x^2+3 x+2 , & if & x R- -1,-2 -1, & if & x=-2 0, & if & x=-1 array . Now, we have to check the continuity aligned & at x= & at x=-2,-1 & aligned LHL & = _ h 0 (-2-h)+2 (-2-h)^2+3(-2-h)+2 & = _ h 0 -h 4+h^2+4 h-6-3 h+2 & = _ h 0 -h h^2+h = _ h 0 -1 h+1 =-1 aligned aligned aligned RHL & = _ h 0 (-2+h)+2 (-2+h)^2+3(-2+h)+2 & = _ h 0 h 4+h^2-4 h-6+3 h+2 & = _ h 0 h h^2-h & = _ h 0 1 h-1 =-1 LHL & = RHL =f(-2) aligned It is continuous at x=-2 Now, check for x=-1 aligned LHL & = _ h 0 (-1-h