SOLUTION: For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.

Question 1209860: For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = \frac{\sqrt{|x|}}{x} + x + \frac{1}{x^2}

Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
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).

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