Let f be a function satisfying ( f(x)=frac{a^{x}}{a^{x}+sqrt{a}}=g_{a}(x)(a>0) )

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.

Explore the function f(x) = a^x / (a^x + sqrt(a)), its behavior as x approaches infinity and negative infinity, and its domain and range. Understand its derivative using the quotient rule and its monotonicity. Suitable for advanced high school students and beyond.

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