Part 1) Recall that to evaluate a function at a given value, we substitute the variable with the given value.
Then:
[tex]f(g(2))=5g(2)+3.[/tex]Now, evaluating g(x) at x=2 we get:
[tex]g(2)=\frac{2+4}{2}.[/tex]Simplifying the above result we get:
[tex]\begin{gathered} g(2)=\frac{6}{2}, \\ g(2)=3. \end{gathered}[/tex]Substituting the above result in the first equation we get:
[tex]f(g(2))=5*3+3.[/tex]Simplifying the above result we get:
[tex]\begin{gathered} f(g(2))=15+3, \\ f(g(2))=18. \end{gathered}[/tex]Answer Part 1:
[tex]f(g(2))=18.[/tex]Part 2) Analogously to part1 we get that:
[tex]g(f(2))=\frac{f(2)+4}{2}.[/tex]Evaluating f(x) at x=2 we get:
[tex]f(2)=5*2+3.[/tex]Simplifying the above result we get:
[tex]\begin{gathered} f(2)=10+3, \\ f(2)=13. \end{gathered}[/tex]Then:
[tex]g(f(2))=\frac{13+4}{2}.[/tex]Simplifying the above result we get:
[tex]g(f(2))=\frac{17}{2}.[/tex]Answer Part2:
[tex]g(f(2))=\frac{17}{2}.[/tex]