Sure! Let's solve the given question step-by-step.
We are given two functions:
[tex]\[ r(x) = 2\sqrt{x} \][/tex]
[tex]\[ s(x) = \sqrt{x} \][/tex]
### Part 1: Calculating [tex]\((rs)(4)\)[/tex]
The notation [tex]\((rs)(4)\)[/tex] represents the product of the functions [tex]\(r(x)\)[/tex] and [tex]\(s(x)\)[/tex] evaluated at [tex]\(x = 4\)[/tex].
1. Evaluate [tex]\(r(4)\)[/tex]:
[tex]\[ r(4) = 2\sqrt{4} \][/tex]
2. Evaluate [tex]\(s(4)\)[/tex]:
[tex]\[ s(4) = \sqrt{4} \][/tex]
3. Calculate the product [tex]\((rs)(4)\)[/tex]:
[tex]\[ (rs)(4) = r(4) \cdot s(4) \][/tex]
From the above evaluations:
[tex]\[ r(4) = 2 \times \sqrt{4} = 2 \times 2 = 4 \][/tex]
[tex]\[ s(4) = \sqrt{4} = 2 \][/tex]
Thus:
[tex]\[ (rs)(4) = 4 \times 2 = 8 \][/tex]
### Part 2: Calculating [tex]\(\left(\frac{r}{s}\right)(3)\)[/tex]
The notation [tex]\(\left(\frac{r}{s}\right)(3)\)[/tex] represents the quotient of the functions [tex]\(r(x)\)[/tex] and [tex]\(s(x)\)[/tex] evaluated at [tex]\(x = 3\)[/tex].
1. Evaluate [tex]\(r(3)\)[/tex]:
[tex]\[ r(3) = 2\sqrt{3} \][/tex]
2. Evaluate [tex]\(s(3)\)[/tex]:
[tex]\[ s(3) = \sqrt{3} \][/tex]
3. Calculate the quotient [tex]\(\left(\frac{r}{s}\right)(3)\)[/tex]:
[tex]\[ \left(\frac{r}{s}\right)(3) = \frac{r(3)}{s(3)} \][/tex]
From the above evaluations:
[tex]\[ r(3) = 2 \times \sqrt{3} \][/tex]
[tex]\[ s(3) = \sqrt{3} \][/tex]
Thus:
[tex]\[ \left(\frac{r}{s}\right)(3) = \frac{2\sqrt{3}}{\sqrt{3}} = 2 \][/tex]
### Summary
- [tex]\((rs)(4) = 8\)[/tex]
- [tex]\(\left(\frac{r}{s}\right)(3) = 2\)[/tex]
So, we have the final results:
[tex]\[
\begin{array}{c}
(r s)(4)=8 \\
\left(\frac{r}{s}\right)(3)=2
\end{array}
\][/tex]
