Question

given the function $f(x)=(x + 2)^3$, which graph represents the inverse of the function?

Explanation:

Step1: Find inverse of $f(x)$

Start with $y=(x+2)^3$. Swap $x$ and $y$: $x=(y+2)^3$. Solve for $y$:
$$\sqrt[3]{x} = y + 2 \implies y = \sqrt[3]{x} - 2$$

Step2: Identify key points of inverse

• When $x=0$: $y=\sqrt[3]{0}-2=-2$, so point $(0,-2)$
• When $x=1$: $y=\sqrt[3]{1}-2=-1$, so point $(1,-1)$
• When $x=-8$: $y=\sqrt[3]{-8}-2=-2-2=-4$, so point $(-8,-4)$
• When $x=8$: $y=\sqrt[3]{8}-2=2-2=0$, so point $(8,0)$

Step3: Match points to graphs

The inverse function $y=\sqrt[3]{x}-2$ is a cube root function shifted down 2 units. It passes through $(0,-2)$, $(1,-1)$, $(-8,-4)$, $(8,0)$, which matches the first graph (top-most option).

Answer:

The top-most graph (first option, passing through $(0,-2)$ and $(8,0)$) represents the inverse function.