Question

Need some help with my last problem for the night.

Determine the domain of tbe function (f o g)(x) where f(x) = 3x-1/x-4 and g(x) = x+1/x

Answer 1

We are given

f(x)= $\frac{3x-1}{x-4}$

g(x)=$\frac{x+1}{x}$

\to find f o g, replace all the x in f(x) with g(x)

$f o g = \frac{3(\frac{x+1}{x})-1}{(\frac{x+1}{x})-4}$

= Simplify by taking least common denominator and reciprocal.

Taking the least common denominator, we get

$\frac{(\frac{2x+3}{x})}{(\frac{-3x+1}{x})}$

Taking reciprocal, we get

$\frac{2x+3}{x}\ast \frac{x}{-3x+1}$

Simplifying the expression, we get

$f o g =\frac{2x+3}{-3x+1}$

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

$-3x+1=0$

$x=\frac{1}{3}$

It means, domain is all real numbers except $x=\frac{1}{3}$

Domain of g(x) is all real numbers except $x=0$

combined domain is $\left ( -\infty ,0 \right )\cup \left ( 0,\frac{1}{3} \right )\cup \left (  \frac{1}{3},\infty \right )$