Questions: Refer to the functions r, p, and q. Find the function (r/q)(x) and write the domain in interval notation. r(x) = 4x p(x) = x^2 + 2x q(x) = sqrt(8-x) Part: 0 / 2 Part 1 of 2 (r/q)(x) =

Refer to the functions r, p, and q. Find the function (r/q)(x) and write the domain in interval notation.

r(x) = 4x
p(x) = x^2 + 2x
q(x) = sqrt(8-x)

Part: 0 / 2

Part 1 of 2

(r/q)(x) =

Transcript text: Refer to the functions $r, p$, and $q$. Find the function $\left(\frac{r}{q}\right)(x)$ and write the domain in interval notation. \[ r(x)=4 x \quad p(x)=x^{2}+2 x \quad q(x)=\sqrt{8-x} \] Part: $0 / 2$ Part 1 of 2 \[ \left(\frac{r}{q}\right)(x)= \] $\square$

Solution

Solution Steps

Step 1: Define the Function $\left(\frac{r}{q}\right)(x)$

To find the function $\left(\frac{r}{q}\right)(x)$, we need to divide the function $r(x)$ by the function $q(x)$.

Given: \[ r(x) = 4x \] \[ q(x) = \sqrt{8-x} \]

The function $\left(\frac{r}{q}\right)(x)$ is defined as: \[ \left(\frac{r}{q}\right)(x) = \frac{r(x)}{q(x)} = \frac{4x}{\sqrt{8-x}} \]

Step 2: Determine the Domain of $\left(\frac{r}{q}\right)(x)$

The domain of $\left(\frac{r}{q}\right)(x)$ is determined by the restrictions on $q(x)$, since division by zero is undefined and the square root function requires non-negative arguments.

Square Root Restriction: The expression under the square root, $8-x$, must be non-negative: \[ 8-x \geq 0 \implies x \leq 8 \]
Denominator Restriction: The denominator $\sqrt{8-x}$ cannot be zero: \[ \sqrt{8-x} \neq 0 \implies 8-x \neq 0 \implies x \neq 8 \]

Combining these conditions, the domain of $\left(\frac{r}{q}\right)(x)$ is: \[ x \leq 8 \quad \text{and} \quad x \neq 8 \]

Thus, the domain in interval notation is: \[ (-\infty, 8) \]