Given the function $f(x) = \frac{a^x}{a^x + \sqrt{a}}$, where $a > 0$, we need to analyze and interpret this function, which is defined as $g_a(x)$.

Understanding the Function $f(x)$:

Let's break down the function step by step.

1. Definition: The function is given by: $f(x) = \frac{a^x}{a^x + \sqrt{a}}$ Here, $a > 0$ is a constant, and $x$ is the variable.

2. Domain: Since $a^x$ and $\sqrt{a}$ are well-defined for all real numbers $x$, the domain of $f(x)$ is $x \in \mathbb{R}$.

3. Behavior at $x = 0$: $f(0) = \frac{a^0}{a^0 + \sqrt{a}} = \frac{1}{1 + \sqrt{a}}$

4. Behavior as $x \to \infty$: As $x$ increases, $a^x$ becomes very large compared to $\sqrt{a}$, so: $\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{a^x}{a^x + \sqrt{a}} = \frac{a^x}{a^x \left(1 + \frac{\sqrt{a}}{a^x}\right)} = \frac{1}{1 + 0} = 1$

5. Behavior as $x \to -\infty$: As $x$ decreases, $a^x$ becomes very small compared to $\sqrt{a}$, so: $\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \frac{a^x}{a^x + \sqrt{a}} = \frac{0}{0 + \sqrt{a}} = 0$

Summary of Properties:

• $f(x)$ starts from 0 as $x \to -\infty$.
• $f(x)$ increases and approaches 1 as $x \to \infty$.
• The function is continuous and strictly increasing.

Important Points:

• Range: The range of $f(x)$ is $(0, 1)$ for all $x \in \mathbb{R}$.
• Monotonicity: Since $f(x)$ is increasing, it is a one-to-one function.
• Derivative: To understand the rate of change, you could compute $f'(x)$ using the quotient rule.

Would you like to explore the derivative or the inverse of this function, or have any other questions?

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Here are 8 related questions to consider:

1. What is the derivative of $f(x) = \frac{a^x}{a^x + \sqrt{a}}$?
2. How does the function $f(x)$ behave if $a = 1$?
3. What is the inverse function of $f(x)$?
4. How does the function change if $x$ is replaced by $-x$?
5. What is the integral of $f(x)$ over $(-\infty, \infty)$?
6. Can you prove that $f(x)$ is strictly increasing?
7. What happens to $f(x)$ when $a < 1$?
8. How can you express $f(x)$ as a composition of simpler functions?

Tip: For functions involving exponential terms, understanding their asymptotic behavior at $\pm \infty$ helps to visualize the function's range and limits.