Answer:
The inverse of g(x) is [tex]g^{-1}(x)=\frac{2x}{1-x}[/tex]
The domain of g(x) is (-∞,-2)U(-2,∞)
The domain of f(x) is (-∞,-1)U(-1,1)U(1,∞)
Step-by-step explanation:
Find the inverse of g(x).
[tex]y=\frac{x}{x+2}[/tex]
Solve for x
[tex]x=y(x+2)[/tex]
[tex]x=xy+2y[/tex]
[tex]x(1-y)=2y[/tex]
[tex]x=\frac{2y}{1-y}[/tex]
[tex]g^{-1}(x)=\frac{2x}{1-x}[/tex]
Find the domain of g(x)
[tex]g(x)=\frac{x}{x+2}[/tex]
Numerator and denominator have domain
(-∞,∞).
However, g(x) is undefined if the denominator is 0. Hence, x=-2 must be taken out of the domain. So, the domain of g(x) is
(-∞,-2)U(-2,∞).
Find the domain of f(x)
[tex]f(x)=\frac{1}{x^{2} -1}[/tex]
The numerator and denominator have the domain
(-∞,∞)
However, f(x) is undefined if the denominator is 0. This rules out +1 and -1. So, the domain of f(x) is
(-∞,-1)U(-1,1)U(1,∞)
I am not sure what you mean for (ii).
