Question

Help please thank you trying to learn before my upcoming test I have missed school due to health issues

Answer 1

Given:

$\begin{gathered} f(x)=\frac{1}{x-5} \\ g(x)=\frac{3}{x}+5 \end{gathered}$

To Determine:

$\begin{gathered} (fog)(x) \\ (gof)(x) \end{gathered}$

Solution

$\begin{gathered} (fog)(x)=f(g(x)) \\ (fog)(x)=f(\frac{3}{x}+5) \\ (fog)(x)=\frac{1}{\frac{3}{x}+5-5} \\ (fog)(x)=\frac{1}{\frac{3}{x}} \\ (fog)(x)=1\div\frac{3}{x} \\ (fog)(x)=1\times\frac{x}{3} \\ (fog)(x)=\frac{x}{3} \end{gathered}$$\begin{gathered} (gof)(x)=g(f(x) \\ (gof)(x)=g(\frac{1}{x-5}) \\ (gof)(x)=\frac{3}{\frac{1}{x-5}}+5 \\ (gof)(x)=3\div(\frac{1}{x-5})+5 \\ (gof)(x)=3\times(x-5)+5 \\ (gof)(x)=3x-15+5 \\ (gof)(x)=3x-10 \end{gathered}$

Hence,

$\begin{gathered} (fog)(x)=\frac{x}{3} \\ (gof)(x)=3x-10 \end{gathered}$