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The question asks to find the domain and range for three functions:

1. $f(x) = \sqrt{2x + 3}$
2. $g(x) = \frac{1}{4x - 1}$
3. $h(x) = \sqrt{(x + 1)^{-1}}$

Solutions:

1. For $f(x) = \sqrt{2x + 3}$:

   • Domain: Since square roots require the argument to be non-negative, we need $2x + 3 \geq 0$. Solving this gives $x \geq -\frac{3}{2}$. Therefore, the domain is: $x \in \left[ -\frac{3}{2}, \infty \right)$
   • Range: The output of a square root function is non-negative, so the range is: $f(x) \geq 0 \quad \text{or} \quad f(x) \in [0, \infty)$

2. For $g(x) = \frac{1}{4x - 1}$:

   • Domain: The function is undefined when the denominator is zero, i.e., $4x - 1 = 0$ implies $x = \frac{1}{4}$. Thus, the domain is all real numbers except $x = \frac{1}{4}$: $x \in (-\infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty)$
   • Range: Since this is a rational function, $g(x)$ can take any real value except 0, which is asymptotically approached but never reached. Hence, the range is: $g(x) \in (-\infty, 0) \cup (0, \infty)$

3. For $h(x) = \sqrt{(x + 1)^{-1}}$:

   • Domain: The argument of the square root must be non-negative, i.e., $(x + 1)^{-1} \geq 0$. This implies that $x + 1 > 0$, so $x > -1$. Therefore, the domain is: $x \in (-1, \infty)$
   • Range: Since the square root function is non-negative, the range of this function is: $h(x) \in [0, \infty)$

Would you like more details or clarifications? Here are five related questions to further explore these concepts:

1. How do we determine the domain of a function involving a square root?
2. What does the behavior of rational functions near vertical asymptotes tell us about their range?
3. How would the domain of a function change if the expression under the square root was negative?
4. Can you explain why $g(x)$ never equals zero?
5. How do the properties of inverse powers, like in $h(x) = \sqrt{(x+1)^{-1}}$, influence the function's domain?

Tip: When finding the domain of functions involving square roots, always start by solving the inequality inside the square root for non-negative values.