Check if the function ||f(x)=-(x^2)+1 if x<3 | 2 if x=3 | x-11 if x>3|| is limited on x=a and sketch the graph.  

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embizze | High School Teacher | (Level 2) Educator Emeritus

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We are given a piecewise function:

`f(x)={ [-x^2+1,x<3],[2,x=3],[x-11,x>3]]`

The limit as x approaches 3 exists and the limit is -8. `lim_(x->3^(-))=-8,lim_(x->3^(+))=-8` . Since the value of the function at x=3 is 2 which does not equal the limit, there is a nonremovable discontinuity at x=3. The function is continuous everywhere else.


The graph:

The function is not differentiable at x=3, and wouldn't be if f(3)=-8 since the derivative from the left and right do not agree. The function is differentiable everywhere else.