The domain of a function is the set of all values that go into a function. Our function is
[tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)}=\frac{x^{2}-6x+9}{x-3}[/tex]
To find the domain, first let's rewrite the numerator as a product of binomials
[tex]x^2-6x+9=(x-3)^2[/tex]
Then, our function is
[tex](\frac{f}{g})(x)=\frac{x^{2}-6x+9}{x-3}=\frac{(x-3)^2}{x-3}=x-3[/tex]
Our function is a line equation, and a line is defined everywhere, therefore, the domain of our function is
[tex](-\infty,\infty)[/tex]
