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Let's solve the given problems for each function: \( f(x) \), \( g(x) \), and \( h(x) \).

### Given Functions:
1. \( f(x) = | -14 + 8x | \)
2. \( g(x) = \sqrt{x - 2} + 3 \)
3. \( h(x) = \frac{3 + x^2}{x + 7} \)

### Problems:
We need to find \( f\left(\frac{1}{4}\right) \), \( g(5) \), and \( h(5) \).

### Solutions:
#### 1. Finding \( f\left(\frac{1}{4}\right) \):
   Substitute \( x = \frac{1}{4} \) into \( f(x) \):
   \[
   f\left(\frac{1}{4}\right) = \left| -14 + 8 \cdot \frac{1}{4} \right|
   \]
   Simplify inside the absolute value:
   \[
   = \left| -14 + 2 \right| = \left| -12 \right| = 12
   \]
   So, \( f\left(\frac{1}{4}\right) = 12 \).

#### 2. Finding \( g(5) \):
   Substitute \( x = 5 \) into \( g(x) \):
   \[
   g(5) = \sqrt{5 - 2} + 3
   \]
   Simplify inside the square root:
   \[
   = \sqrt{3} + 3
   \]
   So, \( g(5) = \sqrt{3} + 3 \).

#### 3. Finding \( h(5) \):
   Substitute \( x = 5 \) into \( h(x) \):
   \[
   h(5) = \frac{3 + 5^2}{5 + 7}
   \]
   Simplify:
   \[
   = \frac{3 + 25}{12} = \frac{28}{12} = \frac{7}{3}
   \]
   So, \( h(5) = \frac{7}{3} \).

### Final Answers:
- \( f\left(\frac{1}{4}\right) = 12 \)
- \( g(5) = \sqrt{3} + 3 \)
- \( h(5) = \frac{7}{3} \)

Would you like a more detailed explanation on any of these steps?

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Here are some related questions to explore further:
1. How does the absolute value function affect the shape of \( f(x) \)?
2. What are the domain and range of \( g(x) = \sqrt{x - 2} + 3 \)?
3. How would \( h(x) \) behave as \( x \to -7 \)?
4. Can \( f(x) \) have any negative outputs?
5. How does the square root in \( g(x) \) affect the rate of increase compared to a linear function?

**Tip:** When working with functions involving absolute values or square roots, always consider restrictions on the domain to ensure valid inputs.

Find f(1/4), g(5), and h(5) for the given functions: f(x) = | -14 + 8x |, g(x) = √(x - 2) + 3, h(x) = (3 + x^2) / (x + 7). Simplify your answers as much as possible.

This math problem involves evaluating functions with absolute value, square roots, and rational expressions. Learn how to compute f(1/4) = 12, g(5) = √3 + 3, and h(5) = 7/3 with detailed steps. Suitable for high school students in Grades 9-12.

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