Math Problem Statement

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

Solution

The function given in the image is h(x)=xx2+1h(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)h(x) is defined for all real values of xx. This is because the denominator x2+1\sqrt{x^2 + 1} is always positive (since x2+1≥1x^2 + 1 \geq 1) and never zero. Therefore, the function has no restrictions on the domain: Domain: (−∞,∞)\text{Domain: } (-\infty, \infty)

2. Behavior as xx Approaches Infinity

As xx approaches infinity or negative infinity, the function behaves like: h(x)≈x∣x∣=sgn(x)h(x) \approx \frac{x}{|x|} = \text{sgn}(x) where sgn(x)\text{sgn}(x) is the sign function:

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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Calculus
Limits
Derivative Analysis

Formulas

h(x) = x / sqrt(x^2 + 1)
Limit as x approaches infinity: lim(x->∞) h(x) = 1
Derivative: h'(x) = 1 / (x^2 + 1)^(3/2)

Theorems

Limit Theorems
Derivative Theorems

Suitable Grade Level

Grades 10-12

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