https://file.aiquickdraw.com/mathbot/file/1725410132928lddw3s3d.jpg

The function given in the image is \( h(x) = \frac{x}{\sqrt{x^2 + 1}} \), along with a graph representing it. Let's analyze the function and the graph step by step.

### 1. **Domain of the Function**
The function \( h(x) \) is defined for all real values of \( x \). This is because the denominator \( \sqrt{x^2 + 1} \) is always positive (since \( x^2 + 1 \geq 1 \)) and never zero. Therefore, the function has no restrictions on the domain:
\[
\text{Domain: } (-\infty, \infty)
\]

### 2. **Behavior as \( x \) Approaches Infinity**
As \( x \) approaches infinity or negative infinity, the function behaves like:
\[
h(x) \approx \frac{x}{|x|} = \text{sgn}(x)
\]
where \( \text{sgn}(x) \) is the sign function:
\[
\text{sgn}(x) = \begin{cases} 
1 & \text{if } x > 0, \\
-1 & \text{if } x < 0, \\
0 & \text{if } x = 0.
\end{cases}
\]
Thus, as \( x \to \infty \), \( h(x) \to 1 \), and as \( x \to -\infty \), \( h(x) \to -1 \).

### 3. **Derivative and Slope Analysis**
The derivative of \( h(x) \) helps determine the slope and critical points:
\[
h'(x) = \frac{\sqrt{x^2 + 1} - \frac{x^2}{\sqrt{x^2 + 1}}}{x^2 + 1}
\]
Simplifying gives:
\[
h'(x) = \frac{1}{(x^2 + 1)^{3/2}}
\]
Since \( h'(x) \) is always positive, the function \( h(x) \) is increasing for all \( x \).

### 4. **Range of the Function**
The range is between the horizontal asymptotes at \( y = -1 \) and \( y = 1 \), with the function smoothly transitioning from -1 to 1 as \( x \) increases.

### 5. **Graph Characteristics**
The graph provided seems to illustrate:
- A horizontal asymptote at \( y = 1 \) as \( x \) approaches \( \infty \).
- A horizontal asymptote at \( y = -1 \) as \( x \) approaches \( -\infty \).
- Smooth transition between these values, with the slope decreasing as \( |x| \) increases.

The graph matches these characteristics, confirming the analysis.

Would you like a more detailed explanation on any of these points, or do you have any specific questions?

Here are 5 related questions you might consider:
1. What would be the effect on the graph if we modify the function to \( h(x) = \frac{x}{\sqrt{x^2 + k}} \), where \( k \) is a positive constant?
2. How does the graph of \( h(x) = \frac{x}{\sqrt{x^2 + 1}} \) compare to \( g(x) = \frac{x}{\sqrt{x^2 + 2}} \)?
3. How can we determine the concavity of the function \( h(x) \)?
4. What is the significance of the derivative \( h'(x) \) in terms of the graph's steepness?
5. How would the function behave if we replaced the square root in the denominator with a cubic root?

**Tip:** Understanding the asymptotic behavior of functions is crucial in graphing and analyzing limits.

Analyze the function h(x) = x / sqrt(x^2 + 1) and describe its graph.

Explore the graph of h(x) = x / sqrt(x^2 + 1) with detailed analysis on its domain, behavior as x approaches infinity, and derivative-based slope. Suitable for Grades 10-12.

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