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) =

Solution Steps

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: $$\left(\frac{r}{q}\right)(x) = \frac{r(x)}{q(x)} = \frac{4x}{\sqrt{8-x}}$$

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.

1. Square Root Restriction: The expression under the square root, $8-x$, must be non-negative: $$8-x \geq 0 \implies x \leq 8$$

2. 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: $$(-\infty, 8)$$

Final Answer