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The problem asks to find the domains of functions \( f(x) = \sqrt{x + 3} \) and \( g(x) = \frac{x}{2} \), as well as the domain of the composition \( f \circ g \).

Let’s break it down:

### (a) Domain of \( f(x) = \sqrt{x + 3} \)
For the square root function to be real, the expression inside the square root must be non-negative:
\[
x + 3 \geq 0
\]
Solving this inequality:
\[
x \geq -3
\]
Thus, the domain of \( f \) is \( [-3, \infty) \).

### (b) Domain of \( g(x) = \frac{x}{2} \)
The function \( g(x) = \frac{x}{2} \) is defined for all real values of \( x \). So, the domain of \( g \) is \( (-\infty, \infty) \).

### (c) Domain of \( f \circ g \)
The composition \( f \circ g \) means \( f(g(x)) = \sqrt{g(x) + 3} \).

Since \( g(x) = \frac{x}{2} \), we have:
\[
f(g(x)) = \sqrt{\frac{x}{2} + 3}
\]
For this expression to be defined, the inside of the square root must be non-negative:
\[
\frac{x}{2} + 3 \geq 0
\]
Solving this inequality:
\[
\frac{x}{2} \geq -3 \quad \Rightarrow \quad x \geq -6
\]
Thus, the domain of \( f \circ g \) is \( [-6, \infty) \).

### Summary:
- Domain of \( f \): \( [-3, \infty) \)
- Domain of \( g \): \( (-\infty, \infty) \)
- Domain of \( f \circ g \): \( [-6, \infty) \)

Would you like any further details or clarifications?

### 5 Related Questions:
1. What is the difference between the domain and the range of a function?
2. How do you find the domain of a rational function?
3. Can a function have more than one domain depending on transformations?
4. How would the domain change if \( f(x) = \sqrt{2x + 1} \)?
5. How does the domain change for compositions of more complex functions?

### Tip:
For compositions of functions, always ensure the output of the inner function lies within the domain of the outer function.

Determine the domains of the following: (a) Domain of f(x) = sqrt(x + 3), (b) Domain of g(x) = x/2, and (c) Domain of the composition f(g(x)).

Learn how to find the domains of functions f(x) = sqrt(x + 3) and g(x) = x/2, along with their composition. This problem involves domain restrictions for square root functions and the composition of functions. Suitable for high school students in Grades 10-12.

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