Question 1209856
Let's analyze the function f(x) = 1/√(x² + 1) - 1/x to determine its properties.

**1. Natural Domain:**

* **√(x² + 1):** x² + 1 is always positive for any real number x, so the square root is defined for all real numbers.
* **1/x:** This is undefined when x = 0.

Therefore, the natural domain of f(x) 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) = 1/√((-x)² + 1) - 1/(-x)
f(-x) = 1/√(x² + 1) + 1/x

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

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

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).

f(x) = (x² + 1)^(-1/2) - x^(-1)

Now, find the derivative:

f'(x) = (-1/2)(x² + 1)^(-3/2)(2x) + x^(-2)
f'(x) = -x / (x² + 1)^(3/2) + 1/x²

To analyze this, we need to find when f'(x) > 0 (increasing) and when f'(x) < 0 (decreasing).

f'(x) = (-x * x²) + (x² + 1)^(3/2) / (x² * (x² + 1)^(3/2))
f'(x) = (-x³ + (x² + 1)^(3/2)) / (x² * (x² + 1)^(3/2))

* **For x > 0:**
    * x² * (x² + 1)^(3/2) is always positive.
    * We need to compare x³ and (x² + 1)^(3/2).
    * When x is small, (x² + 1)^(3/2) will be larger than x³.
    * When x is large, (x² + 1)^(3/2) will be larger than x³.
    * Therefore, f'(x) is likely positive for x>0, thus increasing.
* **For x < 0:**
    * x² * (x² + 1)^(3/2) is always positive.
    * -x³ is positive.
    * (x² + 1)^(3/2) is positive.
    * Therefore f'(x) is likely positive for x<0, thus increasing.

It appears the function is increasing over its entire domain.

**4. Invertibility:**

* If a function is strictly increasing or strictly decreasing, it is invertible.
* Since the function appears to be strictly increasing over its domain, it is **invertible**.

**Conclusion:**

* **Domain:** (-∞, 0) U (0, ∞)
* **Neither even nor odd**
* **Invertible:** Yes
* **Increasing:** Yes