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Question 1209865: 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) = x + 2 \lfloor x \rfloor + sqrt{x} - \frac{1}{|x|}
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's analyze the function f(x) = x + 2⌊x⌋ + √x - 1/|x| to determine its properties.
**1. Natural Domain:**
* **√x:** x ≥ 0
* **1/|x|:** x ≠ 0
* Combining these, the domain is x > 0, or (0, ∞).
**2. Even or Odd:**
* Since the domain is (0, ∞), the function is not symmetric about the y-axis or the origin. Therefore, it cannot be even or odd.
**3. Increasing or Decreasing:**
* **f(x) = x + 2⌊x⌋ + √x - 1/x** (since x > 0, |x| = x)
* Let's analyze the behavior of each term:
* **x:** Increasing
* **2⌊x⌋:** Increasing (step function)
* **√x:** Increasing
* **-1/x:** Increasing
* Since all terms are increasing over the domain (0, ∞), the function f(x) is **increasing** over its domain.
**4. Invertibility:**
* Since the function is strictly increasing over its domain, it is **invertible**.
**Summary:**
* **Domain:** (0, ∞)
* **Neither even nor odd**
* **Invertible:** Yes
* **Increasing:** Yes
* **Decreasing:** No
Answer by ikleyn(52810)  (Show Source):
You can put this solution on YOUR website! .
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) = x + 2 \lfloor x \rfloor + sqrt{x} - \frac{1}{|x|}
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In your post, some parts of the formula are either incorrect or unreadable.
In his post, @CPhill incorrectly interprets your formula, so, his solution is IRRELEVANT.
I just several times pointed to you that this forum accepts messages only in plain text format.
Therefore, copy-paste of formulas from other sources does not work.
You should print your messages manually, using your keyboard.
Otherwise, your posts will go directly to garbage bins - - -
- - - because they do not fit for any other purposes.
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