We are given the rational function $r(x) = \frac{2x}{x + 5}$, and we need to evaluate the following values:

(a) $r(0)$:

$r(0) = \frac{2(0)}{0 + 5} = \frac{0}{5} = 0$

(b) $r(-1)$:

$r(-1) = \frac{2(-1)}{-1 + 5} = \frac{-2}{4} = -\frac{1}{2}$

(c) $r(-3)$:

$r(-3) = \frac{2(-3)}{-3 + 5} = \frac{-6}{2} = -3$

(d) $r\left(\frac{1}{2}\right)$:

$r\left(\frac{1}{2}\right) = \frac{2\left(\frac{1}{2}\right)}{\frac{1}{2} + 5} = \frac{1}{\frac{11}{2}} = \frac{1 \times 2}{11} = \frac{2}{11}$

(e) $r(0.1)$:

$r(0.1) = \frac{2(0.1)}{0.1 + 5} = \frac{0.2}{5.1} \approx 0.0392$

(f) $r(10,000)$:

$r(10,000) = \frac{2(10,000)}{10,000 + 5} = \frac{20,000}{10,005} \approx 1.999$

Final Results:

• (a) $r(0) = 0$
• (b) $r(-1) = -\frac{1}{2}$
• (c) $r(-3) = -3$
• (d) $r\left(\frac{1}{2}\right) = \frac{2}{11}$
• (e) $r(0.1) \approx 0.0392$
• (f) $r(10,000) \approx 1.999$

Would you like more details or have any questions?

Here are five follow-up questions:

1. What is the domain of the function $r(x) = \frac{2x}{x + 5}$?
2. How does the function behave as $x \to \infty$?
3. Can you determine the vertical asymptote of the function?
4. What happens to $r(x)$ as $x \to -5$?
5. How would you find the inverse of $r(x)$?

Tip: When solving rational functions, always check for any values of $x$ that make the denominator zero, as these are not in the domain of the function.