ANSWER:
[tex]\begin{equation*} (f\circ g)(x)=\frac{3}{x+1} \end{equation*}[/tex][tex]Domain:(-\infty,-1)\cup(-1,\infty)[/tex]
EXPLANATION:
Given:
[tex]\begin{gathered} f(x)=\frac{1}{x} \\ g(x)=\frac{x+1}{3} \end{gathered}[/tex]
To find (f o g)(x), we have to substitute x in f(x) with (x + 1)/3 and simplify as seen below;
[tex]\begin{gathered} (f\circ g)(x)=f(g(x))=\frac{1}{\frac{x+1}{3}}=1\div\frac{x+1}{3}=1*\frac{3}{x+1}=\frac{3}{x+1} \\ \therefore(f\circ g)(x)=\frac{3}{x+1} \end{gathered}[/tex]
Recall that the domain of a function is the set of input values for which a function is defined.
So to determine the domain of the stated function, we have to equate the denominator to zero and solve for x as seen below;
[tex]\begin{gathered} x+1=0 \\ x=-1 \end{gathered}[/tex]
We can see that for the given function to be defined, x must not be equal to -1, so we can go ahead and write the domain of the function in interval notation as;
[tex]Domain:(-\infty,-1)\cup(-1,\infty)[/tex]
