To find the inverse of the function $(f \circ g)(x)$, we first need to determine $(f \circ g)(x)$ and then find its inverse.

Step 1: Determine $(f \circ g)(x)$

The composition $(f \circ g)(x)$ means we substitute $g(x)$ into $f(x)$: $(f \circ g)(x) = f(g(x)) = f\left(\frac{4x - 5}{2x + 1}\right)$

Since $f(x) = 3x + 4$, we substitute $\frac{4x - 5}{2x + 1}$ for $x$ in $f$: $f\left(\frac{4x - 5}{2x + 1}\right) = 3 \cdot \frac{4x - 5}{2x + 1} + 4$

Expanding this: $= \frac{12x - 15}{2x + 1} + 4$

To simplify, we rewrite 4 as $\frac{4(2x + 1)}{2x + 1}$: $= \frac{12x - 15 + 4(2x + 1)}{2x + 1}$

Expanding $4(2x + 1)$: $= \frac{12x - 15 + 8x + 4}{2x + 1}$ $= \frac{20x - 11}{2x + 1}$

Thus, $(f \circ g)(x) = \frac{20x - 11}{2x + 1}$.

Step 2: Find the Inverse of $(f \circ g)(x)$

Now we find $(f \circ g)^{-1}(x)$ by setting $(f \circ g)(x) = y$ and solving for $x$: $y = \frac{20x - 11}{2x + 1}$

1. Multiply both sides by $2x + 1$: $y(2x + 1) = 20x - 11$
2. Expand and rearrange terms to isolate $x$: $2xy + y = 20x - 11$ $2xy - 20x = -y - 11$ $x(2y - 20) = -y - 11$ $x = \frac{-y - 11}{2y - 20}$

So, the inverse function is: $(f \circ g)^{-1}(x) = \frac{-x - 11}{2x - 20}$

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Would you like further explanation on any of these steps or have specific questions?

1. What are the general steps for finding the inverse of a composite function?
2. How do you handle restrictions in the domain when finding an inverse?
3. Could there be multiple inverses for a composite function?
4. What is the significance of non-permissible values in the denominator?
5. How does finding an inverse change for functions with more complex expressions?

Tip: To verify if your inverse function is correct, substitute it back into the original function to see if it returns the input $x$.