The function f(x) = |x| + x ^ 2 + x ^ 4 will be?

To analyze the function \( f(x) = |x| + x^2 + x^4 \), let's consider its behavior across different domains of \( x \).

### 1. **Analyzing the Function on Different Domains**

The absolute value function \( |x| \) affects the behavior of \( f(x) \) depending on whether \( x \) is non-negative or negative:

- **For \( x \geq 0 \):**
  \[
  |x| = x
  \]
  Thus, the function simplifies to:
  \[
  f(x) = x + x^2 + x^4
  \]

- **For \( x < 0 \):**
  \[
  |x| = -x
  \]
  Hence, the function becomes:
  \[
  f(x) = -x + x^2 + x^4
  \]

### 2. **Behavior of \( f(x) \) Based on the Sign of \( x \)**

#### Case 1: \( x \geq 0 \)
When \( x \geq 0 \), the function is:
\[
f(x) = x + x^2 + x^4
\]
All the terms \( x, x^2, x^4 \) are non-negative for \( x \geq 0 \), so \( f(x) \geq 0 \). This function is continuous and differentiable for all \( x \geq 0 \).

#### Case 2: \( x < 0 \)
When \( x < 0 \), the function is:
\[
f(x) = -x + x^2 + x^4
\]
Since \( x < 0 \), \( -x > 0 \). The terms \( x^2 \) and \( x^4 \) are also positive, meaning that \( f(x) > 0 \) for \( x < 0 \) as well. Thus, \( f(x) \) remains continuous and differentiable for all \( x \).

### 3. **Key Characteristics of the Function \( f(x) = |x| + x^2 + x^4 \)**

- **Continuity**: The function \( f(x) \) is continuous everywhere because \( |x|, x^2, \) and \( x^4 \) are continuous functions.
- **Differentiability**: \( f(x) \) is differentiable everywhere. Though the absolute value function \( |x| \) has a non-differentiable point at \( x = 0 \), the combination \( f(x) = |x| + x^2 + x^4 \) remains differentiable at \( x = 0 \) due to the smoothness added by the \( x^2 \) and \( x^4 \) terms.

### 4. **Graph and Monotonicity**

- **For \( x \geq 0 \)**: \( f(x) = x + x^2 + x^4 \) is strictly increasing since each term \( x, x^2, x^4 \) is increasing.
  
- **For \( x < 0 \)**: \( f(x) = -x + x^2 + x^4 \). Even though \( -x \) is positive and decreasing, \( x^2 + x^4 \) increases faster as \( x \) moves away from zero. Hence, \( f(x) \) also increases.

### 5. **Conclusion**

The function \( f(x) = |x| + x^2 + x^4 \) is:
- **Continuous** and **differentiable** for all \( x \).
- **Strictly increasing** for all \( x \) due to the dominating terms \( x^2 \) and \( x^4 \).

### Summary
\( f(x) = |x| + x^2 + x^4 \) is a smooth, strictly increasing function that is positive for all real numbers.

Would you like to see the detailed graph of this function or have any specific questions about its properties? 

### Additional Questions:

1. How does the graph of \( f(x) = |x| + x^2 + x^4 \) look for different ranges of \( x \)?
2. What are the derivatives of \( f(x) = |x| + x^2 + x^4 \) for \( x \geq 0 \) and \( x < 0 \)?
3. How would the function change if one of the terms were removed, such as \( x^4 \)?
4. Can we determine the concavity of \( f(x) \) over its domain?
5. What is the limit of \( f(x) \) as \( x \) approaches infinity or negative infinity?

**Tip:** When analyzing functions involving absolute values, consider breaking down the function into cases for positive and negative \( x \) to understand its behavior fully.

Explore the function f(x) = |x| + x^2 + x^4 and analyze its properties across different domains of x. Understand its continuity, differentiability, and monotonicity characteristics. Suitable for advanced high school students and beyond.

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