Question

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

Explanation:

Step1: Recall Cube Root Function Behavior

The parent cube root function \( y = \sqrt[3]{x} \) is increasing (as \( x \) increases, \( y \) increases) because its derivative \( y'=\frac{1}{3x^{2/3}} \) is positive for all \( x
eq0 \) and defined at \( x = 0 \) (limit as \( x\to0 \) is \(+\infty\)). To make a cube root function decreasing, we need to reflect it over the \( x \)-axis (multiply by \(- 1\)), which changes the sign of the derivative (making it negative).

Step2: Analyze Each Option

• Option 1: \( f(x)=\sqrt[3]{x - 8} \)

This is a horizontal shift of \( y=\sqrt[3]{x} \) (shift right 8 units). Shifting horizontally does not change the increasing/decreasing behavior. The function is still increasing (derivative \( \frac{1}{3(x - 8)^{2/3}}>0 \) for \( x
eq8 \), increasing at \( x = 8 \)).

• Option 2: \( f(x)=\sqrt[3]{x}-5 \)

This is a vertical shift of \( y=\sqrt[3]{x} \) (shift down 5 units). Vertical shifts do not affect the increasing/decreasing behavior. The function is increasing (derivative \( \frac{1}{3x^{2/3}}>0 \) for \( x
eq0 \), increasing at \( x = 0 \)).

• Option 3: \( f(x)=\sqrt[3]{-(5 - x)}=\sqrt[3]{x - 5} \)

Simplify: \( -(5 - x)=x - 5 \), so this is \( \sqrt[3]{x - 5} \), a horizontal shift of \( y=\sqrt[3]{x} \) (shift right 5 units). Still increasing (same reasoning as Option 1).

• Option 4: \( f(x)=-\sqrt[3]{x}+5 \)

This is the parent function \( y=\sqrt[3]{x} \) reflected over the \( x \)-axis (multiplied by \(-1\)) and then shifted up 5 units. The reflection over the \( x \)-axis changes the function to \( y =-\sqrt[3]{x} \), whose derivative is \( -\frac{1}{3x^{2/3}}<0 \) for all \( x
eq0 \) (and decreasing at \( x = 0 \) as the limit of the derivative as \( x\to0 \) is \(-\infty\)). The vertical shift does not affect the increasing/decreasing behavior (only changes the \( y \)-intercept). So this function is decreasing as \( x \) increases.

Answer:

\( f(x)=-\sqrt[3]{x}+5 \) (the fourth option: \( \boldsymbol{f(x)=-\sqrt[3]{x}+5} \))