To analyze the function \( y = \log_2(x^2 + 1) \), let's delve into several key aspects:

1. **Domain**:
   \[
   \text{The domain of the function is the set of all } x \text{ for which the argument of the logarithm is positive.}
   \]
   \[
   x^2 + 1 > 0 \quad \text{for all real numbers } x
   \]
   \[
   \text{Thus, the domain is } (-\infty, \infty).
   \]

2. **Range**:
   \[
   \text{The range of the function is determined by the values that } y \text{ can take.}
   \]
   \[
   x^2 \geq 0 \implies x^2 + 1 \geq 1
   \]
   \[
   \log_2(x^2 + 1) \geq \log_2(1) = 0
   \]
   \[
   \text{As } x^2 \to \infty, \; x^2 + 1 \to \infty \implies \log_2(x^2 + 1) \to \infty
   \]
   \[
   \text{Thus, the range is } [0, \infty).
   \]

3. **Behavior and Symmetry**:
   \[
   \text{Since } x^2 \text{ is an even function, } x^2 + 1 \text{ is also an even function.}
   \]
   \[
   \log_2(x^2 + 1) \text{ is also even, which means it is symmetric about the y-axis.}
   \]
   \[
   \log_2(-x^2 + 1) = \log_2(x^2 + 1)
   \]

4. **Intercepts**:
   - **y-intercept**:
     \[
     \text{Set } x = 0 \implies y = \log_2(0^2 + 1) = \log_2(1) = 0
     \]
     \[
     \text{The y-intercept is } (0, 0).
     \]
   - **x-intercepts**:
     \[
     \text{There are no x-intercepts because } \log_2(x^2 + 1) \geq 0 \text{ for all } x.
     \]

5. **Asymptotes**:
   \[
   \text{There are no vertical asymptotes since the logarithm's argument is always positive.}
   \]
   \[
   \text{There are no horizontal asymptotes since } \log_2(x^2 + 1) \text{ increases without bound as } x \to \infty.
   \]

6. **First Derivative (to find critical points and analyze increasing/decreasing behavior)**:
   \[
   y' = \frac{d}{dx} \log_2(x^2 + 1)
   \]
   \[
   y' = \frac{1}{\ln(2)} \cdot \frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{(x^2 + 1) \ln(2)}
   \]
   \[
   \text{Setting } y' = 0 \text{ gives } x = 0.
   \]
   \[
   \text{For } x > 0, y' > 0 \text{ and for } x < 0, y' < 0.
   \]
   \[
   \text{Thus, the function is increasing for } x > 0 \text{ and decreasing for } x < 0.
   \]

7. **Second Derivative (to determine concavity)**:
   \[
   y'' = \frac{d}{dx} \left( \frac{2x}{(x^2 + 1) \ln(2)} \right)
   \]
   \[
   y'' = \frac{2((x^2 + 1) \ln(2) - 2x^2)}{((x^2 + 1) \ln(2))^2} = \frac{2(\ln(2) - x^2 \ln(2))}{((x^2 + 1) \ln(2))^2}
   \]
   \[
   y'' = \frac{2(1 - x^2)}{(x^2 + 1)^2 \ln(2)}
   \]
   \[
   \text{Setting } y'' = 0 \text{ gives } x^2 = 1 \implies x = \pm 1.
   \]

   \[
   \text{For } |x| < 1, y'' > 0 \implies \text{concave up}.
   \]
   \[
   \text{For } |x| > 1, y'' < 0 \implies \text{concave down}.
   \]

Would you like any additional details or have any questions about this analysis?

Here are 8 related questions to expand on this information:
1. How do you find the domain of a logarithmic function?
2. What are the properties of logarithms?
3. Can you explain how to determine the range of a function?
4. How do you find the intercepts of a function?
5. What are the steps to find the first derivative of a function?
6. How does the first derivative test help in finding local maxima and minima?
7. What does the second derivative tell us about the concavity of a function?
8. How can you sketch the graph of a logarithmic function given its key characteristics?

**Tip:** Understanding the properties of logarithms can greatly simplify solving and analyzing logarithmic functions.

Explore the domain, range, symmetry, intercepts, asymptotes, derivatives, and concavity of the logarithmic function y = log2(x^2 + 1). This detailed analysis covers key concepts in logarithmic functions and calculus, suitable for advanced high school students and college-level learners.

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