To check for continuity at the edges of each piece, you need to consider the limit as
approaches the edges. For example,
![g(x)=\begin{cases}2x+5&\text{for }x\le-3\\x^2-10&\text{for }x>-3\end{cases}](https://tex.z-dn.net/?f=g%28x%29%3D%5Cbegin%7Bcases%7D2x%2B5%26%5Ctext%7Bfor%20%7Dx%5Cle-3%5C%5Cx%5E2-10%26%5Ctext%7Bfor%20%7Dx%3E-3%5Cend%7Bcases%7D)
has two pieces,
and
, both of which are continuous by themselves on the provided intervals. In order for
to be continuous everywhere, we need to have
![\displaystyle\lim_{x\to-3^-}g(x)=\lim_{x\to-3^+}g(x)=g(-3)](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bx%5Cto-3%5E-%7Dg%28x%29%3D%5Clim_%7Bx%5Cto-3%5E%2B%7Dg%28x%29%3Dg%28-3%29)
By definition of
, we have
, and the limits are
![\displaystyle\lim_{x\to-3^-}g(x)=\lim_{x\to-3}(2x+5)=-1](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bx%5Cto-3%5E-%7Dg%28x%29%3D%5Clim_%7Bx%5Cto-3%7D%282x%2B5%29%3D-1)
![\displaystyle\lim_{x\to-3^+}g(x)=\lim_{x\to-3}(x^2-10)=-1](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bx%5Cto-3%5E%2B%7Dg%28x%29%3D%5Clim_%7Bx%5Cto-3%7D%28x%5E2-10%29%3D-1)
The limits match, so
is continuous.
For the others: Each of the individual pieces of
are continuous functions on their domains, so you just need to check the value of each piece at the edge of each subinterval.
B. 11
Although eleven is a prime number, it's two factors (1 and 11) are odd integers
32-x=0; with x equaling the amount descended or -32.
-3^3=9
18-9+3•2
9+3•2
12•2
24