Given the functions f(x) and g(x) defined by:
[tex]\begin{gathered} f(x)=\sqrt{\frac{2x}{3}} \\ \\ g(x)=\sqrt{x^2+15} \end{gathered}[/tex]
To find the composite function evaluated at 1, we use the definition of composite functions:
[tex](f\circ g)(x)=f(g(x))[/tex]
Then, for x = 1, we find g(1):
[tex]\begin{gathered} g(1)=\sqrt{1^2+15}=\sqrt{16} \\ \\ \Rightarrow g(1)=4 \end{gathered}[/tex]
Now, we use this result in f(x):
[tex]\begin{gathered} f(4)=\sqrt{\frac{2\cdot4}{3}}=\sqrt{\frac{8}{3}} \\ \\ \Rightarrow f(4)\approx1.63 \end{gathered}[/tex]
Finally:
[tex]\begin{gathered} (f\circ g)(1)=f(g(1))=f(4) \\ \\ \therefore(f\circ g)(1)=1.63 \end{gathered}[/tex]
