To find four ordered pairs on the graph of the rational function $f(x) = \frac{1}{x} + 9$, we substitute various $x$-values into the function and compute the corresponding $y$-values.

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Step-by-step process:

1. Choose values of $x$: Select a mix of positive and negative values, and avoid $x = 0$ since the function is undefined at $x = 0$.

2. Substitute into $f(x) = \frac{1}{x} + 9$: Calculate $y$ for each chosen $x$.

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Ordered pairs:

Here are the calculations for specific values of $x$:

1. For $x = 1$:
   $f(1) = \frac{1}{1} + 9 = 1 + 9 = 10$
   Ordered pair: $(1, 10)$

2. For $x = -1$:
   $f(-1) = \frac{1}{-1} + 9 = -1 + 9 = 8$
   Ordered pair: $(-1, 8)$

3. For $x = 2$:
   $f(2) = \frac{1}{2} + 9 = 0.5 + 9 = 9.5$
   Ordered pair: $(2, 9.5)$

4. For $x = -2$:
   $f(-2) = \frac{1}{-2} + 9 = -0.5 + 9 = 8.5$
   Ordered pair: $(-2, 8.5)$

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Final Answer:

The four ordered pairs are:
$(1, 10), \, (-1, 8), \, (2, 9.5), \, (-2, 8.5)$

Would you like a graph or further explanation?

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Related Questions:

1. What is the domain and range of $f(x) = \frac{1}{x} + 9$?
2. How does the vertical asymptote at $x = 0$ affect the graph of $f(x)$?
3. What happens to $f(x)$ as $x \to \infty$ or $x \to -\infty$?
4. Can you identify the horizontal asymptote of $f(x)$?
5. How would the graph of $f(x) = \frac{1}{x} + 9$ change if the $+9$ were replaced with $+k$?

Tip: Always check the domain of rational functions to avoid dividing by zero.