To solve the problem $f \circ g$, which is the composition of two functions $f(x) = x^2 - 4x$ and $g(x) = \frac{1}{x+1}$, where $x \neq -1$, we need to substitute $g(x)$ into $f(x)$.

Step-by-step solution:

1. The composition $f \circ g$ means we substitute $g(x)$ into $f(x)$. $(f \circ g)(x) = f(g(x))$

2. Since $f(x) = x^2 - 4x$, we substitute $g(x) = \frac{1}{x+1}$ into $f(x)$: $f(g(x)) = \left( \frac{1}{x+1} \right)^2 - 4 \left( \frac{1}{x+1} \right)$

3. Simplify each term:

   • First, $\left( \frac{1}{x+1} \right)^2 = \frac{1}{(x+1)^2}$
   • Second, $4 \left( \frac{1}{x+1} \right) = \frac{4}{x+1}$

   So the expression becomes: $f(g(x)) = \frac{1}{(x+1)^2} - \frac{4}{x+1}$

4. To simplify further, find a common denominator:

   • The common denominator is $(x+1)^2$, so we rewrite the second term: $\frac{4}{x+1} = \frac{4(x+1)}{(x+1)^2} = \frac{4x + 4}{(x+1)^2}$

5. Now, subtract the two fractions: $f(g(x)) = \frac{1 - (4x + 4)}{(x+1)^2} = \frac{1 - 4x - 4}{(x+1)^2} = \frac{-4x - 3}{(x+1)^2}$

Thus, the composition $(f \circ g)(x)$ is: $(f \circ g)(x) = \frac{-4x - 3}{(x+1)^2}$

Would you like more details on any of the steps or have any further questions?

Here are 5 related questions:

1. How do we determine the domain of the composite function $f \circ g$?
2. Can we reverse the composition and find $g \circ f$?
3. What is the significance of finding the composition of two functions?
4. How can we graph the composite function $(f \circ g)(x)$?
5. What happens if $g(x)$ is undefined, and how does that affect the composition?

Tip: When composing functions, always check the domain of each function involved to avoid undefined expressions.