Question

the function $g(x)$ is a transformation of the cube root parent function, $f(x)=\sqrt3{x}$. what function is $g(x)$?
a. $g(x)=\frac{1}{4}\sqrt3{x}$
b. $g(x)=\sqrt3{x + 4}$
c. $g(x)=\sqrt3{x}+4$
d. $g(x)=4\sqrt3{x}$

Explanation:

Step1: Identify parent function points

For $f(x)=\sqrt[3]{x}$, when $x=1$, $f(1)=\sqrt[3]{1}=1$; when $x=8$, $f(8)=\sqrt[3]{8}=2$.

Step2: Match g(x) points to transformation

From the graph, at $x=1$, $g(1)=4$; at $x=8$, $g(8)=8$. This shows $g(x)$ is a vertical stretch of $f(x)$.

Step3: Verify stretch factor

A vertical stretch by factor $k$ follows $g(x)=k\cdot f(x)$. Since $g(1)=4\cdot f(1)=4\cdot1=4$, $k=4$. So $g(x)=4\sqrt[3]{x}$.

Step4: Eliminate other options

• Option A: $\frac{1}{4}\sqrt[3]{x}$ would shrink the graph, which does not match.
• Option B: $\sqrt[3]{x+4}$ is a horizontal shift left, which does not match the vertical stretch.
• Option C: $\sqrt[3]{x}+4$ is a vertical shift up, which would not scale the output values proportionally.

Answer:

D. $g(x) = 4\sqrt[3]{x}$