A function is defined on with its graph described below
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A function q is defined on (−6,6) with its graph described below.
• For −6<x<−2, the graph is the line segment joining (−6,−4) to (−2,0).
• For −2≤x<1, the graph is the logistic curve y=1+e−(x+1)3−2.
• At x=1, there is an open circle at (1,1) and a filled square at (1,−3).
• For 1<x<3, the graph is the constant y=−3.
• For x>3, the graph is y=x−35+2.
Parts:
(a) List every x–value in (−6,6) at which q is not continuous.
(b) For each discontinuity identified in part (a), provide a complete limit–based justification of its classification (removable, jump, or infinite), including any necessary one–sided limits and function values.
(c) Construct an explicit piecewise definition of q that incorporates the formulas and isolated points described above. Your definition must account for domain restrictions and individual plotted points.
(d) The rational branch for x>3 can be generalized to y=x−a5+2 where a>0. Explain how varying the parameter a affects the location of the discontinuity and the continuity of the extended function on (−6,6).