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 \[ \left(\frac{r}{q}\right)(x)= \] $\square$
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Solution

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Solution Steps

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

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

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

The function (rq)(x)\left(\frac{r}{q}\right)(x) is defined as: (rq)(x)=r(x)q(x)=4x8x \left(\frac{r}{q}\right)(x) = \frac{r(x)}{q(x)} = \frac{4x}{\sqrt{8-x}}

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

The domain of (rq)(x)\left(\frac{r}{q}\right)(x) is determined by the restrictions on q(x)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, 8x8-x, must be non-negative: 8x0    x8 8-x \geq 0 \implies x \leq 8

  2. Denominator Restriction: The denominator 8x\sqrt{8-x} cannot be zero: 8x0    8x0    x8 \sqrt{8-x} \neq 0 \implies 8-x \neq 0 \implies x \neq 8

Combining these conditions, the domain of (rq)(x)\left(\frac{r}{q}\right)(x) is: x8andx8 x \leq 8 \quad \text{and} \quad x \neq 8

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

Final Answer

The function (rq)(x)\left(\frac{r}{q}\right)(x) is: (rq)(x)=4x8x \left(\frac{r}{q}\right)(x) = \frac{4x}{\sqrt{8-x}}

The domain of (rq)(x)\left(\frac{r}{q}\right)(x) is: (,8) \boxed{(-\infty, 8)}

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