SOLUTION
From the original function
[tex]f(x)=\sqrt[3]{x}[/tex]
a vertical stretch means multiplying f(x) by a number greater than 1. So a vertical stretch by a factor of 4 means multiplying f(x) by 4 so we have
[tex]4\sqrt[3]{x}[/tex]
A horizontal shrink means dividing the graph by that factor, bringing it closer to the y-axis, so the x-coordinate should be divided by that factor.
Hence a horizontal shift by a factor of 1/3 means dividing x by 1/3, which is same as multiplying x by 3, so we have
[tex]\begin{gathered} =4\sqrt[3]{x\times3} \\ g(x)=4\sqrt[3]{3x} \end{gathered}[/tex]
Hence Mike is correct.
Also let us compare their graphs
This is Mike's graph
And this is Lucy's graph
You can see that Mike's graph is more stretched (by a factor of 4) than that of Lucy (by a factor of 3)
Hence Mike is correct
