Let g(x) = ln(x +√(x^2 + 1)).
(a) Find the domain of g.

To find the domain of the function \( g(x) = \ln(x + \sqrt{x^2 + 1}) \), we need to determine the set of values for \( x \) for which the argument of the logarithm is positive, since the logarithmic function is only defined for positive real numbers.

### Step 1: Determine the argument of the logarithm
The argument of the logarithm in \( g(x) \) is:
\[
x + \sqrt{x^2 + 1}
\]

### Step 2: Determine when \( x + \sqrt{x^2 + 1} > 0 \)
Let's analyze this expression:

1. **Check the expression \( \sqrt{x^2 + 1} \):**
   - The square root function is always non-negative, i.e., \( \sqrt{x^2 + 1} \geq 0 \).
   - For all real \( x \), \( x^2 + 1 \) is always greater than or equal to 1, hence \( \sqrt{x^2 + 1} \geq 1 \).

2. **Check the whole expression \( x + \sqrt{x^2 + 1} \):**
   - For \( x \geq 0 \), since \( \sqrt{x^2 + 1} > x \) (because \( \sqrt{x^2 + 1} = \sqrt{x^2 + 1} \geq |x| \geq x \)), we have \( x + \sqrt{x^2 + 1} > 0 \).
   - For \( x < 0 \), note that \( \sqrt{x^2 + 1} > |x| \), which implies \( \sqrt{x^2 + 1} > -x \). Therefore, \( x + \sqrt{x^2 + 1} > 0 \).

Hence, for all real numbers \( x \), \( x + \sqrt{x^2 + 1} > 0 \).

### Step 3: Conclusion
Since the expression \( x + \sqrt{x^2 + 1} \) is always positive for all real values of \( x \), the domain of \( g(x) \) is:

\[
\boxed{(-\infty, \infty)}
\]

This means \( g(x) \) is defined for all real numbers.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How would you find the derivative of \( g(x) = \ln(x + \sqrt{x^2 + 1}) \)?
2. What is the range of \( g(x) = \ln(x + \sqrt{x^2 + 1}) \)?
3. Can you find the inverse of \( g(x) \)?
4. How does the graph of \( g(x) \) look like?
5. What is the behavior of \( g(x) \) as \( x \to \infty \) and \( x \to -\infty \)?

**Tip:** When dealing with logarithmic functions, always ensure that the argument inside the logarithm is positive to determine the domain correctly.

Let g(x) = ln(x + √(x^2 + 1)). (a) Find the domain of g.

Explore how to find the domain of the logarithmic function g(x) = ln(x + √(x^2 + 1)). This solution covers key concepts like the properties of logarithms and square root functions, providing a clear explanation for students in advanced high school or undergraduate math courses.

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