Question 1209860
Let's analyze the function f(x) = (√|x| / x) + x + (1/x²) to determine its properties.

**1. Natural Domain:**

* The square root requires |x| ≥ 0, which is true for all x.
* The terms 1/x and 1/x² require x ≠ 0.
* Therefore, the natural domain is all real numbers except 0, or (-∞, 0) U (0, ∞).

**2. Even or Odd:**

* **Even:** f(-x) = f(x)
* **Odd:** f(-x) = -f(x)

Let's test f(-x):

f(-x) = (√|-x| / -x) - x + (1/(-x)²)
f(-x) = (√|x| / -x) - x + (1/x²)
f(-x) = -(√|x| / x) - x + (1/x²)

Now, let's see if f(-x) = f(x) or f(-x) = -f(x):

* f(x) = (√|x| / x) + x + (1/x²)
* -f(x) = -(√|x| / x) - x - (1/x²)

Comparing:

* f(-x) ≠ f(x) (Not even)
* f(-x) ≠ -f(x) (Not odd)

Therefore, the function is **neither even nor odd**.

**3. Increasing or Decreasing:**

* To determine if the function is increasing or decreasing, we need to analyze its derivative, f'(x).

Let's break down f(x) into cases:

* **For x > 0:** f(x) = (√x / x) + x + (1/x²) = x^(-1/2) + x + x^(-2)
* **For x < 0:** f(x) = (√(-x) / x) + x + (1/x²) = -(-x)^(-1/2) + x + x^(-2)

Now, find the derivatives:

* **For x > 0:**
    * f'(x) = (-1/2)x^(-3/2) + 1 - 2x^(-3) = (-1/2x√x) + 1 - (2/x³)
* **For x < 0:**
    * f'(x) = (-1/2)(-x)^(-3/2) + 1 - 2x^(-3) = (1/2(-x)√-x) + 1 - (2/x³)

Analyzing the derivatives is complex and doesn't reveal a simple answer. It's difficult to make a general statement about increasing or decreasing behavior across the entire domain.
Therefore, we cannot easily state if the function is increasing or decreasing. A graphing calculator would be helpful to determine this.

**4. Invertibility:**

* A function is invertible if it is one-to-one (passes the horizontal line test).
* Since the function is not strictly increasing or decreasing across its entire domain, it is **not invertible**. Also, because it is not even or odd, it is likely not invertible.

**Conclusion:**

* **Domain:** (-∞, 0) U (0, ∞)
* **Neither even nor odd**
* **Invertible:** No
* **Increasing/Decreasing:** Difficult to determine without further analysis (graphing is recommended).