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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.

Find f(1/2), g(8), and h(2) for the functions f(x) = | -7 + 8x |, g(x) = √(x - 5) - 5, and h(x) = (x^2 + 4)/(x + 4). Simplify your answers.

Learn how to evaluate functions such as f(x) = | -7 + 8x |, g(x) = √(x - 5) - 5, and h(x) = (x^2 + 4)/(x + 4). This tutorial covers key mathematical concepts including absolute value, square root, and rational functions, providing step-by-step solutions to find f(1/2), g(8), and h(2). Suitable for students in Grades 9-11.

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