Let's analyze and simplify the function:

### Given Function:
\[
g(x) = \frac{x^2}{x^2 + 4}
\]

### 1. **Simplification**:
This function is already simplified, but let's consider the behavior of the function as \( x \) becomes very large and at specific values.

### 2. **Domain**:
The function is defined for all real numbers \( x \), since the denominator \( x^2 + 4 \) is always positive (never zero).

### 3. **Asymptotic Behavior**:
- **Horizontal Asymptote**: As \( x \) approaches \( \pm \infty \), the function behaves like:
  \[
  \lim_{x \to \pm \infty} \frac{x^2}{x^2 + 4} = \lim_{x \to \pm \infty} \frac{1}{1 + \frac{4}{x^2}} = 1
  \]
  So, \( y = 1 \) is a horizontal asymptote.

### 4. **First Derivative (Rate of Change)**:
To find the derivative \( g'(x) \), use the quotient rule:
\[
g'(x) = \frac{(x^2 + 4) \cdot \frac{d}{dx}(x^2) - x^2 \cdot \frac{d}{dx}(x^2 + 4)}{(x^2 + 4)^2}
\]

Let's compute the derivative step by step:

- The derivative of the numerator \( x^2 \) is \( 2x \).
- The derivative of the denominator \( x^2 + 4 \) is \( 2x \).

So:
\[
g'(x) = \frac{(x^2 + 4)(2x) - x^2(2x)}{(x^2 + 4)^2} = \frac{2x(x^2 + 4 - x^2)}{(x^2 + 4)^2} = \frac{8x}{(x^2 + 4)^2}
\]

### 5. **Critical Points**:
To find the critical points, set \( g'(x) = 0 \):
\[
\frac{8x}{(x^2 + 4)^2} = 0
\]
This occurs when \( 8x = 0 \), so \( x = 0 \).

### 6. **Second Derivative (Concavity)**:
To determine the concavity, you can differentiate \( g'(x) \) to find \( g''(x) \), but the behavior around \( x = 0 \) and \( x \to \pm \infty \) gives a good understanding of the function's shape.

Would you like a more detailed explanation on any of these points or further exploration into the second derivative?

Here are 5 related questions:
1. What is the behavior of \( g(x) \) near \( x = 0 \)?
2. How does the function \( g(x) = \frac{x^2}{x^2 + 4} \) compare to similar rational functions?
3. What happens to \( g(x) \) as \( x \) approaches positive or negative infinity?
4. How does the quotient rule apply to more complex functions?
5. What is the significance of critical points in determining the local maxima or minima of \( g(x) \)?

**Tip:** When differentiating using the quotient rule, always remember: "Derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared."

Explore the analysis of the rational function g(x) = x^2/(x^2 + 4) including simplification, asymptotic behavior, derivatives, and critical points. Understand its behavior as x approaches infinity and at specific values. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation