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The functions given in the problem are:

\[
f(x) = \frac{2 + x^2}{6 + x}, \quad g(x) = \left| -15 - \frac{1}{4}x \right|, \quad h(x) = \sqrt{x + 12} + 5
\]

We need to find the values of \( f(4) \), \( g(12) \), and \( h(-3) \). Let's calculate each step-by-step:

1. **Finding \( f(4) \):**

   Substitute \( x = 4 \) into the function \( f(x) \):

   \[
   f(4) = \frac{2 + 4^2}{6 + 4} = \frac{2 + 16}{10} = \frac{18}{10} = \frac{9}{5}
   \]

2. **Finding \( g(12) \):**

   Substitute \( x = 12 \) into the function \( g(x) \):

   \[
   g(12) = \left| -15 - \frac{1}{4} \times 12 \right| = \left| -15 - 3 \right| = \left| -18 \right| = 18
   \]

3. **Finding \( h(-3) \):**

   Substitute \( x = -3 \) into the function \( h(x) \):

   \[
   h(-3) = \sqrt{-3 + 12} + 5 = \sqrt{9} + 5 = 3 + 5 = 8
   \]

**Simplified Answers:**

- \( f(4) = \frac{9}{5} \)
- \( g(12) = 18 \)
- \( h(-3) = 8 \)

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:

1. What are the domain restrictions for each of these functions?
2. How do you find the inverse of these functions?
3. How would the values change if you altered the input values slightly?
4. Can you sketch the graphs of these functions based on the given definitions?
5. How do absolute values affect the computation of functions like \( g(x) \)?

**Tip:** Always simplify fractions or radicals as much as possible to make your answer clear and concise!

Find f(4), g(12), and h(-3) based on the following functions: f(x) = (2 + x^2) / (6 + x), g(x) = | -15 - (1/4)x |, h(x) = sqrt(x + 12) + 5

Step-by-step guide to finding f(4), g(12), and h(-3) for the given algebraic functions f(x), g(x), and h(x). Learn how to use absolute values, square roots, and basic fraction operations in this algebra problem. Suitable for students in Grades 9-10.

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