Question

This is a question for my Calc 1 class at ASU.

Answer 1

The two pieces are continuous for every $c \in \mathbb{R}$

So, our only concern is to make sure that the pieces "glue" continuously at x=7.

To ensure this, we evaluate both pieces at x=7, and impose that the two values are equal.

The first piece evaluates to

$f_1(7)=7^2-c = 49-c$

The second piece evaluates to

$f_2(7)=7c+1$

So, we want

$7c+1=49-c$

We move all terms involving c to the left hand side, and all the numerical constants on the right hand side:

$8c=48 \iff c=6$