Given the functions f(x) and g(x), as follows:
[tex]f(x)=\frac{1}{x-4}[/tex]and
[tex]g(x)=\frac{5}{x}+4[/tex]Now:
(a) f(g(x)) is obtained as follows:
[tex]\begin{gathered} f(x)=\frac{1}{x-4}\text{ and }g(x)=\frac{5}{x}+4 \\ \Rightarrow f(g(x))=\frac{1}{(\frac{5}{x}+4)-4}\text{ } \\ \Rightarrow f(g(x))=\frac{1}{\frac{5}{x}}=\frac{x}{5} \\ \Rightarrow f(g(x))=\frac{x}{5} \end{gathered}[/tex](b) g(f(x)) is obtained as follows:
[tex]\begin{gathered} f(x)=\frac{1}{x-4}\text{ and }g(x)=\frac{5}{x}+4 \\ \Rightarrow g(f(x))=\frac{5}{\frac{1}{x-4}}+4=5(x-4)+4 \\ \Rightarrow g(f(x))=5(x-4)+4=5x-20+4=5x-16 \\ \Rightarrow g(f(x))=5x-16 \end{gathered}[/tex]