A function is defined piecewise by p(x)= cases x^ 2 -1,&x 1, 4-x,&x>1

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A function

p

is defined piecewise by

p(x)=\begin{cases}x^{2}-1,&x\le 1,\\4-x,&x>1.\end{cases}

Throughout this problem, consider the closed interval

[-2,3]

. (a) Verify that

p

is continuous on

[-2,3]

. Provide complete reasoning, especially at any point where the formula for

p

changes. (b) Calculate

p(-2)

and

p(3)

. (c) Show that the equation

p(x)=2

has at least one solution

c

with

-2<c<3

. Explain your reasoning in clear, logical steps. (d) Determine whether

p(x)=5

must have a solution in

[-2,3]

. Justify your answer fully.