We are given
f(x)= [tex]\frac{3x-1}{x-4}[/tex]
g(x)=[tex]\frac{x+1}{x}[/tex]
\to find f o g, replace all the x in f(x) with g(x)
[tex]f o g = \frac{3(\frac{x+1}{x})-1}{(\frac{x+1}{x})-4}[/tex]
= Simplify by taking least common denominator and reciprocal.
Taking the least common denominator, we get
[tex]\frac{(\frac{2x+3}{x})}{(\frac{-3x+1}{x})}[/tex]
Taking reciprocal, we get
[tex]\frac{2x+3}{x}\ast \frac{x}{-3x+1}[/tex]
Simplifying the expression, we get
[tex]f o g =\frac{2x+3}{-3x+1}[/tex]
To find the domain of a composite function, we need to find the domain of f o g and the first function i.e g(x)
To find the domain of f o g, equate the denominator to 0
[tex]-3x+1=0[/tex]
[tex]x=\frac{1}{3}[/tex]
It means, domain is all real numbers except [tex]x=\frac{1}{3}[/tex]
Domain of g(x) is all real numbers except [tex]x=0[/tex]
combined domain is [tex]\left ( -\infty ,0 \right )\cup \left ( 0,\frac{1}{3} \right )\cup \left ( \frac{1}{3},\infty \right )[/tex]
