Let's solve the given functions step by step. The functions $f(x)$, $g(x)$, and $h(x)$ are defined as:

$f(x) = | -7 + 8x |$ $g(x) = \sqrt{x - 5} - 5$ $h(x) = \frac{x^2 + 4}{x + 4}$

We are tasked with finding the following:

1. $f\left(\frac{1}{2}\right)$
2. $g(8)$
3. $h(2)$

Step 1: Find $f\left(\frac{1}{2}\right)$

$f\left(\frac{1}{2}\right) = \left| -7 + 8\left(\frac{1}{2}\right) \right| = \left| -7 + 4 \right| = \left| -3 \right| = 3$

So, $f\left(\frac{1}{2}\right) = 3$.

Step 2: Find $g(8)$

$g(8) = \sqrt{8 - 5} - 5 = \sqrt{3} - 5$

Thus, $g(8) = \sqrt{3} - 5$, which is approximately $-3.267$ if we simplify it numerically.

Step 3: Find $h(2)$

$h(2) = \frac{2^2 + 4}{2 + 4} = \frac{4 + 4}{6} = \frac{8}{6} = \frac{4}{3}$

So, $h(2) = \frac{4}{3}$.

Final Answers:

1. $f\left(\frac{1}{2}\right) = 3$
2. $g(8) = \sqrt{3} - 5$ (approximately $-3.267$)
3. $h(2) = \frac{4}{3}$

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. What is the value of $f(1)$?
2. Can $g(x)$ ever be a positive value? If so, for what $x$?
3. How does the function $h(x)$ behave as $x \to -4$?
4. What is the domain of $g(x)$?
5. How would $f(x)$ change if the expression inside the absolute value was $-7 - 8x$?

Tip:

Always check the domain of square root and rational functions to avoid undefined expressions.