You have linear functions \(p(x)\) and \(q(x)\). You know \(p(2)=3\), and \(p(q(x))=4x+7\) for all \(x\). Find \(q(-1)\).
+26367 You have linear functions \(p(x)\) and \(q(x)\). You know \(p(2)=3\), and \(p\Big(q(x)\Big)=4x+7\) for all x.
Find \(q(-1)\).
\(\text{Let $p(x)=ax+b$}\)
\(\begin{array}{|rcll|} \hline p(x) &=& ax+b \quad | \quad x=2 \\ p(2) &=& 2a+b \quad | \quad p(2) = 3 \\ 3 &=& 2a+b \\ \mathbf{2a} &=& \mathbf{3-b} \\ \hline \end{array} \begin{array}{|rcll|} \hline p\Big(q(x)\Big) &=& aq(x)+b \quad | \quad p\Big(q(x)\Big) = 4x+7 \\ 4x+7 &=& aq(x)+b \quad | \quad x=-1 \\ 4(-1)+7 &=& aq(-1)+b \\ 3 &=& aq(-1)+b \\ 3-b &=& aq(-1) \quad | \quad * 2 \\ 2(3-b) &=& 2aq(-1) \quad | \quad \mathbf{2a=3-b} \\ 2(3-b) &=& (3-b)q(-1) \quad | :( 3-b) \\ 2 &=& q(-1) \\ \mathbf{q(-1)} &=& \mathbf{2} \\ \hline \end{array}\)
