Answer:
[tex]g(x)=4\sqrt[3]{x}[/tex]
Step-by-step explanation:
We can see that the curve has not been translated horizontally or vertically, nor has it been reflected in either axis. Therefore, g(x) is a stretch:
[tex]y=a\:f(x) \implies f(x) \: \textsf{stretched parallel to the y-axis by a factor of}\:a[/tex]
[tex]y=f(a x) \implies f(x) \: \textsf{stretched parallel to the x-axis by a factor of} \: \dfrac{1}{a}[/tex]
Parent function [tex]f(x)=\sqrt[3]{x}[/tex]
[tex]\implies g(x)=a\:f(x)=a\sqrt[3]{x}[/tex]
[tex]\textsf{or}\quad g(x)=f(ax)=\sqrt[3]{ax}[/tex]
From inspection of the graph:
[tex]g(1)=4[/tex]
Therefore:
[tex]\implies a\sqrt[3]{1}=4[/tex]
[tex]\implies a=4[/tex]
[tex]\implies g(x)=4\sqrt[3]{x}[/tex]
Or:
[tex]\implies \sqrt[3]{a(1)}=4[/tex]
[tex]\implies a=4^3[/tex]
[tex]\implies a=64[/tex]
[tex]\implies g(x)=\sqrt[3]{64x}[/tex]
[tex]\implies g(x)=\sqrt[3]{64}\sqrt[3]{x}[/tex]
[tex]\implies g(x)=4\sqrt[3]{x}[/tex]
Therefore, the translated function is:
[tex]g(x)=4\sqrt[3]{x}[/tex]
