ANSWER
[tex]\begin{gathered} (f\cdot f)(x)=x^4+14x^2+49 \\ (g\cdot g)(x)=\frac{64}{9x^2} \end{gathered}[/tex]
EXPLANATION
We are given the two functions:
[tex]\begin{gathered} f(x)=x^2+7 \\ g(x)=\frac{8}{3x} \end{gathered}[/tex]
To find (f * f)(x), we have to find the product of f(x) with itself.
That is:
[tex](f\cdot f)(x)=f(x)\cdot f(x)[/tex]
Therefore, we have:
[tex]\begin{gathered} (f\cdot f)(x)=(x^2+7)(x^2+7) \\ (f\cdot f)(x)=(x^2)(x^2)+(7)(x^2)+(7)(x^2)+(7)(7) \\ (f\cdot f)(x)=x^4+7x^2+7x^2+49 \\ (f\cdot f)(x)=x^4+14x^2+49 \end{gathered}[/tex]
We apply the same procedure to (g * g)(x):
[tex]\begin{gathered} (g\cdot g)(x)=(\frac{8}{3x})(\frac{8}{3x}) \\ (g\cdot g)(x)=\frac{64}{9x^2} \end{gathered}[/tex]
Those are the answers.
