which cube root function is always decreasing as x increases?\n○ (f(x)=sqrt3{x - 8})\n○ (f(x)=sqrt3{x}-5)\n○…

Explanation:

Step1: Recall cube - root function property

The basic cube - root function is $y = \sqrt[3]{x}$, which is an increasing function. For a cube - root function of the form $y=a\sqrt[3]{x - h}+k$, the sign of $a$ determines the increasing or decreasing behavior. If $a>0$, the function is increasing, and if $a < 0$, the function is decreasing.

Step2: Analyze each option

• Option 1: $f(x)=\sqrt[3]{x - 8}$, here $a = 1>0$, so it is an increasing function.
• Option 2: $f(x)=\sqrt[3]{x}-5$, here $a = 1>0$, so it is an increasing function.
• Option 3: $f(x)=\sqrt[3]{-(5 - x)}=\sqrt[3]{x - 5}$, here $a = 1>0$, so it is an increasing function.
• Option 4: $f(x)=-\sqrt[3]{x}+5$, here $a=- 1<0$, so as $x$ increases, $y$ decreases.

Answer:

$f(x)=-\sqrt[3]{x}+5$