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.

Algebra ->  Functions -> 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.       Log On


   

Question 1209855: 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) = sqrt{10 - x} + \sqrt{x + 10}
Answer by CPhill(1987) About Me  (Show Source):
You can put this solution on YOUR website!
Let's analyze the function f(x) = √(10 - x) + √(x + 10).
**1. Natural Domain:**
* **√(10 - x):** 10 - x ≥ 0 => x ≤ 10
* **√(x + 10):** x + 10 ≥ 0 => x ≥ -10
Therefore, the domain is the intersection of these two conditions: -10 ≤ x ≤ 10, or [-10, 10].
**2. Even or Odd:**
* **Even:** f(-x) = f(x)
* **Odd:** f(-x) = -f(x)
Let's test f(-x):
f(-x) = √(10 - (-x)) + √((-x) + 10)
f(-x) = √(10 + x) + √(10 - x)
Comparing:
* f(-x) = √(10 + x) + √(10 - x)
* f(x) = √(10 - x) + √(10 + x)
Therefore, f(-x) = f(x), and the function is **even**.
Since the function is even, it cannot be odd.
**3. Increasing or Decreasing:**
* To determine if the function is increasing or decreasing, we need to analyze its derivative, f'(x).
f(x) = (10 - x)^(1/2) + (x + 10)^(1/2)
Now, find the derivative:
f'(x) = (1/2)(10 - x)^(-1/2)(-1) + (1/2)(x + 10)^(-1/2)(1)
f'(x) = -1 / (2√(10 - x)) + 1 / (2√(x + 10))
To find when f'(x) > 0 (increasing) and f'(x) < 0 (decreasing), we need to set f'(x) = 0:
-1 / (2√(10 - x)) + 1 / (2√(x + 10)) = 0
1 / (2√(x + 10)) = 1 / (2√(10 - x))
√(x + 10) = √(10 - x)
x + 10 = 10 - x
2x = 0
x = 0
Now, let's test intervals:
* **For -10 ≤ x < 0:**
* Let's test x = -9:
* f'(-9) = -1 / (2√(10 - (-9))) + 1 / (2√(-9 + 10))
* f'(-9) = -1 / (2√19) + 1 / (2√1)
* f'(-9) ≈ -0.114 + 0.5
* f'(-9) > 0 (increasing)
* **For 0 < x ≤ 10:**
* Let's test x = 9:
* f'(9) = -1 / (2√(10 - 9)) + 1 / (2√(9 + 10))
* f'(9) = -1 / (2√1) + 1 / (2√19)
* f'(9) ≈ -0.5 + 0.114
* f'(9) < 0 (decreasing)
Therefore, the function is increasing on [-10, 0] and decreasing on [0, 10].
**4. Invertibility:**
* Since the function is not strictly increasing or decreasing over its entire domain, it is **not invertible**.
**Conclusion:**
* **Domain:** [-10, 10]
* **Even**
* **Invertible:** No
* **Increasing:** [-10, 0]
* **Decreasing:** [0, 10]