To find the natural domain and range of the function $f(x) = \frac{1}{\sqrt{x} + 2}$, let's break it down into steps.

1. Domain:

The domain of a function consists of all the values of $x$ for which the function is defined.

For $f(x) = \frac{1}{\sqrt{x} + 2}$, we have two key considerations:

• Square root condition: The expression under the square root, $x$, must be non-negative. So, $x \geq 0$.
• Denominator condition: The denominator, $\sqrt{x} + 2$, must not be zero. Since $\sqrt{x} \geq 0$ for all $x \geq 0$, $\sqrt{x} + 2 > 0$. Therefore, there are no additional restrictions from this condition.

Thus, the natural domain of $f(x)$ is all $x$ such that $x \geq 0$.

Domain: $[0, \infty)$

2. Range:

The range of a function consists of all possible values that $f(x)$ can take.

To find the range of $f(x)$:

• As $x$ increases, $\sqrt{x}$ increases, causing $\sqrt{x} + 2$ to increase. Since $f(x) = \frac{1}{\sqrt{x} + 2}$, as $\sqrt{x} + 2$ increases, $f(x)$ decreases.
• At $x = 0$, $\sqrt{x} = 0$, so $f(0) = \frac{1}{0 + 2} = \frac{1}{2}$.
• As $x$ approaches infinity, $\sqrt{x}$ approaches infinity, so $\sqrt{x} + 2$ also approaches infinity, making $f(x)$ approach 0.

Thus, the smallest value of $f(x)$ is 0 (approached as $x$ approaches infinity), and the largest value is $\frac{1}{2}$ (when $x = 0$).

Range: $(0, \frac{1}{2}]$

Summary

• Domain: $[0, \infty)$
• Range: $(0, \frac{1}{2}]$

Would you like further details or explanations?

Here are five related questions you might find helpful:

1. How would the domain and range change if the function were $f(x) = \frac{1}{\sqrt{x - 1} + 2}$?
2. What happens to the range if the function was $f(x) = \frac{1}{\sqrt{x} + 1}$ instead?
3. How do you find the domain and range for the function $g(x) = \frac{1}{\sqrt{x^2 - 4}}$?
4. Can you determine the domain and range of $h(x) = \frac{1}{\sqrt{2x + 1} + 3}$?
5. What if the function were $f(x) = \frac{1}{\sqrt{x}}$? How would the domain and range differ?

Tip: Always check for conditions that could make the denominator zero or the expression under a square root negative when finding the domain of a function.