Answer:
[tex]f(x)=\sqrt[3]{x-4} , g(x)=6x^{2}\textrm{ or }f(x)=\sqrt[3]{x},g(x)=6x^{2} -4[/tex]
Step-by-step explanation:
Given:
The function, [tex]H(x)=\sqrt[3]{6x^{2}-4}[/tex]
Solution 1:
Let [tex]f(x)=\sqrt[3]{x}[/tex]
If [tex]f(g(x))=H(x)=\sqrt[3]{6x^{2}-4}[/tex], then,
[tex]\sqrt[3]{g(x)} =\sqrt[3]{6x^{2}-4}\\g(x)=6x^{2}-4[/tex]
Solution 2:
Let [tex]f(x)=\sqrt[3]{x-4}[/tex]. Then,
[tex]f(g(x))=H(x)=\sqrt[3]{6x^{2}-4}\\\sqrt[3]{g(x)-4}=\sqrt[3]{6x^{2}-4} \\g(x)-4=6x^{2}-4\\g(x)=6x^{2}[/tex]
Similarly, there can be many solutions.
