Question 1209854
Let's analyze the function f(x) = x³ - 3x² + 10x - 15.

**1. Natural Domain:**

* Since f(x) is a polynomial, its natural domain is all real numbers, (-∞, ∞).

**2. Even or Odd:**

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

Let's test f(-x):

f(-x) = (-x)³ - 3(-x)² + 10(-x) - 15
f(-x) = -x³ - 3x² - 10x - 15

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

* f(x) = x³ - 3x² + 10x - 15
* -f(x) = -x³ + 3x² - 10x + 15

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) = 3x² - 6x + 10

To find when f'(x) > 0 (increasing) and when f'(x) < 0 (decreasing), we need to analyze the discriminant of the quadratic f'(x):

Discriminant (Δ) = b² - 4ac
Δ = (-6)² - 4(3)(10)
Δ = 36 - 120
Δ = -84

Since the discriminant is negative, the quadratic f'(x) has no real roots. Also, since the coefficient of x² in f'(x) is positive (3), the parabola opens upwards. This means f'(x) is always positive for all real numbers x.

Therefore, f'(x) > 0 for all x, and the function f(x) is **always increasing** on its domain.

**4. Invertibility:**

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

**Conclusion:**

* **Domain:** (-∞, ∞)
* **Neither even nor odd**
* **Invertible:** Yes
* **Increasing:** Yes
* **Decreasing:** No