Question

If f(x)=x−1/x and g(x)= 1/x +1, then what are the domain and the range of f(g(x)) ?

Answer 1

Answer:

Step-by-step explanation:

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

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

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

Multiply by the LCM, which is (x+1)

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

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

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

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

$f(g(x)) = -x$

Ok now that we have f(x) g(x) and f(g(x)) it is time to find the domain and range.

The domain is the domain of your input, which is g(x) and the the domain of the f(g(x)). The domain of g(x) is all real number except when x = -1

The domain of f(g(x)) is all real numbers. So there are no restrictions so the overall domain is all real number except for when x = -1

Domain

${x \in \!R | x\neq -1$

OR

$(-\infty, -1) U (-1, \infty)$

Now for the range.

Range:

$(-\infty, 1) U (1, \infty)$