We have the following function:
[tex]f(x)=x^{\frac{1}{3}}[/tex]
And we have to predict what would happen if the function changes to:
[tex]f(x)=(x-12)^{\frac{1}{3}}[/tex]
To predict what will happen in this case, we have that:
1. The first function is called the parent function.
2. The second function is a transformation of the parent function.
3. The given transformation is of the form:
[tex]f(x-h)\rightarrow\text{ f\lparen x\rparen has been translated by h units to the right}[/tex]
4. Then we have that, in this case, we have that the parent function has been translated 12 units to the right since we have:
[tex]\begin{gathered} x^{\frac{1}{3}}\rightarrow(x-12)^{\frac{1}{3}}\text{ The parent function has been translated 12 units to the} \\ \text{ right.} \end{gathered}[/tex]
5. And we can check this if we graph the two functions as follows:
Therefore, in summary, we have that:
The graph will shift to the right 12 units (option C.)
