\[ A(x) = \begin{cases} \frac {1}{3} (x-8) & \text { if } -4 \leq k \leq 2 \\ -2x + 14 & \text { if } 4 < k \leq 6 \end{cases} \]
Graph of \(y = A(x)\).
![[Picture]](example_5-b33fe652a989c805b7125c7de56f6791.svg)
Define \(B\) as follows.
\[ B(w) = \begin{cases} \frac {3}{2} w + 3 & \text { if } -6 \leq k < -2 \\ w & \text { if } 3 \leq k < 5 \end{cases} \]
Graph of \(z = B(w)\).
![[Picture]](example_5-2f3698f63efcc50c153da3073217486f.svg)
To investigate the function \(A \circ B\) we need the range of \(B\) and the domain of \(A\).
The range of \(B\) is \(\left [ \answer {-6}, \answer {0} \right ) \cup \left [ \answer {3}, \answer {5} \right )\).
The domain of \(A\) is \(\left [ \answer {-4}, \answer {2} \right ] \cup \left ( \answer {4}, \answer {6} \right ]\).
Their intersection is \(\left [ \answer {-4}, \answer {0} \right ) \cup \left ( \answer {4}, \answer {5} \right )\).