To determine the best approximation for $\sqrt{80}$ using the function $f(x) = \sqrt{x}$, we can apply the linear approximation formula:

$f(a + h) \approx f(a) + f'(a) \cdot h$

where $a$ is a point near $80$, and $h$ is a small increment or decrement from that point.

1. Identify $a$: A suitable choice for $a$ is $81$, since $\sqrt{81} = 9$.

2. Calculate $f'(x)$: $f'(x) = \frac{1}{2\sqrt{x}}$ Therefore, at $x = 81$: $f'(81) = \frac{1}{2\sqrt{81}} = \frac{1}{18}$

3. Calculate $h$: Here, $h = -1$ (since $80 = 81 - 1$).

Using the linear approximation: $f(80) \approx f(81) + f'(81) \cdot (-1)$

This matches option a: $f(81 + (-1)) \approx f'(81)(-1)$

Therefore, the correct answer is: a. $f(81 + (-1)) \approx f'(81)(-1)$

Would you like further details or clarifications?

Here are some related questions you might consider:

1. How does the linear approximation method work?
2. What are the advantages of using linear approximation?
3. Can you explain the concept of derivatives in more detail?
4. What other functions can be approximated using this method?
5. How does this approximation compare to actual calculations of $\sqrt{80}$?

Tip: Always choose $a$ as a point where the function is easy to evaluate.