Problem
Refer to the functions $r, p$, and $q$. Find the function $\left(\frac{p}{q}\right)(x)$ and write the domain in interval notation. \[ r(x)=4 x \quad p(x)=x^{2}+6 x \quad q(x)=\sqrt{4-x} \] Part: 0 / 2 Part 1 of 2 \[ \left(\frac{p}{q}\right)(x)=\square\left[\begin{array}{ccc} \square^{\square} & \sqrt{\square} & \frac{\square}{\square} \\ \times & 5 \end{array}\right. \]
Solution
Step 1 ：Given the functions \(p(x)=x^{2}+6 x\) and \(q(x)=\sqrt{4-x}\), we can form the function \(\left(\frac{p}{q}\right)(x)\) which is \(\frac{x^{2}+6 x}{\sqrt{4-x}}\).
Step 2 ：To find the domain of this function, we need to ensure that the denominator is not zero and the value under the square root is greater than or equal to zero.
Step 3 ：Setting \(4-x \geq 0\) gives us \(x \leq 4\).
Step 4 ：\(\boxed{(-\infty, 4]}\) is the domain of the function \(\left(\frac{p}{q}\right)(x)\) in interval notation.
