Given r(x)=(2x)/(x+5), find the following.
(a) r(0)= 
(b) r(-1)= 
(c) r(-3)= 
(d) r((1)/(2))= 
(e) r(0.1)= 
(f) r(10,000)=

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.

Given r(x) = (2x)/(x+5), find the following. (a) r(0)= (b) r(-1)= (c) r(-3)= (d) r(1/2)= (e) r(0.1)= (f) r(10,000)=

Learn how to evaluate the rational function r(x) = (2x)/(x+5) for different values such as r(0), r(-1), and more. This problem explores key concepts of function evaluation and rational functions, including asymptotic behavior and domain restrictions. Suitable for students in Grades 10-12.

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