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The problem given is:

If \( f(x) = \sqrt{x^2 + 4} \),
Find:  
a) \( f(-3) \)  
b) \( f(2) \)  
c) \( f(x + b \cdot x) \)

Let's go through each part:

### a) \( f(-3) \)
Substitute \( x = -3 \) into the function:
\[
f(-3) = \sqrt{(-3)^2 + 4} = \sqrt{9 + 4} = \sqrt{13}
\]
So, \( f(-3) = \sqrt{13} \).

### b) \( f(2) \)
Substitute \( x = 2 \) into the function:
\[
f(2) = \sqrt{(2)^2 + 4} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
So, \( f(2) = 2\sqrt{2} \).

### c) \( f(x + b \cdot x) \)
Here, substitute \( x + b \cdot x \) into the function \( f(x) \):
\[
f(x + b \cdot x) = \sqrt{(x + bx)^2 + 4}
\]
Simplify the expression inside the square root:
\[
(x + bx)^2 = x^2(1 + b)^2
\]
Thus, the function becomes:
\[
f(x + b \cdot x) = \sqrt{x^2(1 + b)^2 + 4}
\]

Would you like further clarification on any part? 

Here are five related questions for further practice:
1. How would the expression change if the function was \( f(x) = x^2 + 4 \) instead of \( f(x) = \sqrt{x^2 + 4} \)?
2. What is \( f(0) \) for the given function \( f(x) = \sqrt{x^2 + 4} \)?
3. How does the function behave for very large values of \( x \)?
4. What would the derivative \( f'(x) \) be for \( f(x) = \sqrt{x^2 + 4} \)?
5. How would the function change if the \( +4 \) term were replaced by \( -4 \)?

**Tip:** When working with square root functions, ensure the expression under the root is non-negative to maintain real-valued outputs.

If f(x) = sqrt(x^2 + 4), find: a) f(-3), b) f(2), c) f(x + b * x)

This problem covers function evaluation and substitution for f(x) = sqrt(x^2 + 4). Solutions for specific values of x and transformations involving the variable are included. Suitable for students in Grades 9-11.

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