Answer :
Answer: (-∞,-1) ∪ (0,+∞)
Step-by-step explanation: The representation fog(x) is a representation of composite function, meaning one depends on the other.
In this case, fog(x) means:
fog(x) = f(g(x))
fog(x) = [tex]3(x+\frac{1}{x} )-\frac{1}{x+\frac{1}{x} } -4[/tex]
[tex]fog(x)=3x+\frac{3}{x} -\frac{1}{\frac{x^{2}+x}{x} } -4[/tex]
[tex]fog(x)=3x+\frac{3}{x} -\frac{x}{x^{2}+x} -4[/tex]
[tex]fog(x)=\frac{3x^{2}(x^{2}+x)+3(x^{2}+x)-x-4x(x^{2}+x)}{x(x^{2}+x)}[/tex]
[tex]fog(x)=\frac{3x^{4}+3x^{3}+3x^{2}+3x-x-4x^{3}+4x^{2}}{x(x^{2}+x)}[/tex]
[tex]fog(x)=\frac{3x^{4}-x^{3}-x^{2}+2x}{x(x^{2}+x)}[/tex]
This is the function fog(x).
The domain of a function is all the values the independent variable can assume.
For fog(x), denominator can be zero, so:
[tex]x(x^{2}+x) \neq 0[/tex]
If x = 0, the function doesn't exist.
[tex]x^{2}+x \neq0[/tex]
[tex]x(x+1) \neq0[/tex]
[tex]x+1\neq0[/tex]
[tex]x\neq-1[/tex]
Therefore, the domain of this function is: -∞ < -1 or x > 0