? QUESTION
For the functions $f(x)=\frac{x}{x+1}$ and $g(x)=\frac{1}{x}$, find the composition $f \circ g$ and simplify your answer as much as possible. Write the domain using interval notation.
EXPLANATION

Finding $(f \circ g)(x)$ :

Step 1 ：Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))\), so we have \(f(g(x)) = f\left(\frac{1}{x}\right)\)

Step 2 ：Substitute \(\frac{1}{x}\) into \(f(x)\), we get \(f\left(\frac{1}{x}\right) = \frac{\frac{1}{x}}{\frac{1}{x} + 1}\)

Step 3 ：Multiply the numerator and the denominator by \(x\) to simplify the expression, we get \(f\left(\frac{1}{x}\right) = \frac{1}{1 + x}\)

Step 4 ：So, \(f \circ g = \frac{1}{1 + x}\)

Step 5 ：The domain of this function is all real numbers except \(-1\), because \(x\) cannot be \(-1\) as it would make the denominator zero and the function undefined

Step 6 ：So, the domain in interval notation is \((-\infty, -1) \cup (-1, \infty)\)

Step 7 ：\(\boxed{f \circ g = \frac{1}{1 + x}, \text{Domain} = (-\infty, -1) \cup (-1, \infty)}\)