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Let's break down the given expression step by step. The functions are:

- \( f(x) = \sqrt{9 + x^2} \)
- \( g(x) = 3x + 4 \)

You are asked to evaluate the following expression:

\[
\frac{(f(4) \cdot g(4) + 4)}{(gof(0) - 2f(og(0)))}
\]

### Step 1: Compute \( f(4) \) and \( g(4) \)
1. \( f(4) = \sqrt{9 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
2. \( g(4) = 3(4) + 4 = 12 + 4 = 16 \)

### Step 2: Compute the numerator
The numerator is:

\[
f(4) \cdot g(4) + 4 = 5 \cdot 16 + 4 = 80 + 4 = 84
\]

### Step 3: Compute the denominator

#### Compute \( gof(0) \) (i.e., \( g(f(0)) \))
1. \( f(0) = \sqrt{9 + 0^2} = \sqrt{9} = 3 \)
2. \( g(f(0)) = g(3) = 3(3) + 4 = 9 + 4 = 13 \)

#### Compute \( fog(0) \) (i.e., \( f(g(0)) \))
1. \( g(0) = 3(0) + 4 = 4 \)
2. \( f(g(0)) = f(4) = 5 \)

Now the denominator is:

\[
gof(0) - 2f(og(0)) = 13 - 2(5) = 13 - 10 = 3
\]

### Step 4: Compute the final value

Now we can compute the entire expression:

\[
\frac{84}{3} = 28
\]

Thus, the value of the expression is \( 28 \).

Would you like more details or further clarification?

Here are 5 related questions to explore further:
1. How do you evaluate composite functions like \( gof(x) \) and \( fog(x) \)?
2. What are the properties of square root functions like \( f(x) = \sqrt{9 + x^2} \)?
3. How would the result change if the functions \( f(x) \) and \( g(x) \) were modified?
4. What is the significance of composite functions in calculus?
5. Can you think of other mathematical expressions involving functions that you could analyze similarly?

**Tip:** Always check the domain of the functions to ensure the input values are valid for operations like square roots.

Dadas las funciones f(x) = √(9 + x^2) y g(x) = 3x + 4, determine el valor de (f(4)·g(4) + 4) / (gof(0) - 2fog(0)).

Learn how to evaluate the expression (f(4)·g(4) + 4) / (gof(0) - 2fog(0)) where f(x) = √(9 + x^2) and g(x) = 3x + 4. This problem covers key concepts in algebra and composite functions, suitable for students in Grades 10-12.

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