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The question asks to find the domain of the following functions:

### (a) \( f(x) = \frac{2x + 1}{x^2 + x - 2} \)
We need to find the values of \( x \) for which the denominator is not zero. The domain will be all real values except those that make the denominator zero.

The denominator is:
\[
x^2 + x - 2 = 0
\]
Factoring:
\[
(x - 1)(x + 2) = 0
\]
So, \( x = 1 \) and \( x = -2 \) are the values where the denominator is zero. Thus, the domain of \( f(x) \) is:
\[
\boxed{(-\infty, -2) \cup (-2, 1) \cup (1, \infty)}
\]

### (b) \( g(x) = \frac{\sqrt{x}}{x + 1} \)
For the square root function \( \sqrt{x} \) to be defined, \( x \) must be non-negative, meaning \( x \geq 0 \). Also, the denominator cannot be zero, so \( x + 1 \neq 0 \), which gives \( x \neq -1 \).

Thus, the domain of \( g(x) \) is:
\[
\boxed{[0, \infty) \setminus \{-1\}}
\]
This means all non-negative real numbers excluding \( -1 \).

### (c) \( h(x) = \sqrt{x - 1} + \sqrt{x - 1} \)
This function contains square roots, so the expressions inside the square roots must be non-negative. Both square root terms involve \( x - 1 \), so we must have:
\[
x - 1 \geq 0 \quad \Rightarrow \quad x \geq 1
\]
Thus, the domain of \( h(x) \) is:
\[
\boxed{[1, \infty)}
\]

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Would you like any further details or clarification on these calculations? 

Here are 5 related questions to consider:
1. How do you determine the domain of a rational function in general?
2. What happens when there are square roots in the denominator?
3. What is the domain of \( f(x) = \frac{1}{x^2 + 4} \)?
4. How do restrictions in square roots affect the domain?
5. How would the domain change if the function were \( f(x) = \sqrt{1 - x} \)?

**Tip:** Always check for restrictions such as division by zero or square roots of negative numbers when determining a function's domain.

Find the domain of the following functions:
(a) f(x) = (2x + 1) / (x^2 + x - 2)
(b) g(x) = sqrt(x) / (x + 1)
(c) h(x) = sqrt(x - 1) + sqrt(x - 1)

Explore how to find the domain of functions, including rational expressions and square roots. Learn how to handle restrictions in the denominator and square root expressions with step-by-step explanations. Ideal for students in Grades 9-12.

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