Let's determine which cube root function is always decreasing as [tex]\( x \)[/tex] increases. To identify this, we need to evaluate the derivative of each function and check if it is negative for all [tex]\( x \)[/tex].
### Function 1: [tex]\( f(x) = \sqrt[3]{x-8} \)[/tex]
1. Rewrite the function as [tex]\( f(x) = (x-8)^{1/3} \)[/tex].
2. Find the derivative using the chain rule:
[tex]\[
f'(x) = \frac{1}{3} (x-8)^{-2/3} \cdot 1 = \frac{1}{3} (x-8)^{-2/3}
\][/tex]
3. As [tex]\( x \)[/tex] increases, [tex]\( (x-8) \)[/tex] also increases, making [tex]\( (x-8)^{-2/3} \)[/tex] positive. Thus, [tex]\( f'(x) \)[/tex] is positive, indicating [tex]\( f(x) \)[/tex] is increasing.
### Function 2: [tex]\( f(x) = \sqrt[3]{x} - 5 \)[/tex]
1. Rewrite the function as [tex]\( f(x) = x^{1/3} - 5 \)[/tex].
2. Find the derivative:
[tex]\[
f'(x) = \frac{1}{3} x^{-2/3}
\][/tex]
3. For [tex]\( x > 0 \)[/tex], [tex]\( x^{-2/3} \)[/tex] is positive, hence [tex]\( f'(x) \)[/tex] is positive, indicating [tex]\( f(x) \)[/tex] is increasing.
### Function 3: [tex]\( f(x) = \sqrt[3]{-(5-x)} \)[/tex]
1. Rewrite the function as [tex]\( f(x) = (-(5-x))^{1/3} \)[/tex].
2. Simplify and find the derivative:
[tex]\[
f(x) = -(5-x)^{1/3}
\][/tex]
Derivative using the chain rule:
[tex]\[
f'(x) = -\frac{1}{3} (5-x)^{-2/3} \cdot (-1) = \frac{1}{3} (5-x)^{-2/3}
\][/tex]
3. As [tex]\( x \)[/tex] increases and [tex]\( x < 5 \)[/tex], [tex]\( (5-x)^{-2/3} \)[/tex] is positive, making [tex]\( f'(x) \)[/tex] positive. But near and beyond [tex]\( x = 5 \)[/tex], evaluating the behavior of the expression might involve imaginary numbers, which complicates further analysis.
### Function 4: [tex]\( f(x) = -\sqrt[3]{x} + 5 \)[/tex]
1. Rewrite the function as [tex]\( f(x) = -x^{1/3} + 5 \)[/tex].
2. Find the derivative:
[tex]\[
f'(x) = -\frac{1}{3} x^{-2/3}
\][/tex]
3. For [tex]\( x > 0 \)[/tex], [tex]\( x^{-2/3} \)[/tex] is positive, hence [tex]\( -\frac{1}{3} x^{-2/3} \)[/tex] is negative, indicating [tex]\( f(x) \)[/tex] is decreasing.
Thus the function that is always decreasing as [tex]\( x \)[/tex] increases is [tex]\( f(x) = -\sqrt[3]{x} + 5 \)[/tex].
So, the answer is:
[tex]\[
\boxed{f(x) = -\sqrt[3]{x} + 5}
\][/tex]
