Question

(7).Given f(x) =

Answer 1

Answer:

The inverse of g(x) is $g^{-1}(x)=\frac{2x}{1-x}$

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).

$y=\frac{x}{x+2}$

Solve for x

$x=y(x+2)$

$x=xy+2y$

$x(1-y)=2y$

$x=\frac{2y}{1-y}$

$g^{-1}(x)=\frac{2x}{1-x}$

Find the domain of g(x)

$g(x)=\frac{x}{x+2}$

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)

$f(x)=\frac{1}{x^{2} -1}$

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).