We are given two functions, $f(x) = x + 3$ and $g(x) = 2x - 1$, and asked to find the value of $(f \circ g)^{-1}(6)$, which means the inverse of the composite function $f \circ g$ evaluated at 6.

Here’s the step-by-step process:

Step 1: Composite Function $f \circ g$

The composite function $f \circ g$ means applying $g(x)$ first, then applying $f$ to the result.

$f \circ g (x) = f(g(x))$

Substitute $g(x) = 2x - 1$ into $f(x)$:

$f(g(x)) = f(2x - 1) = (2x - 1) + 3 = 2x + 2$

Thus, the composite function is:

$f \circ g (x) = 2x + 2$

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

Now, we need to find the inverse of $f \circ g(x)$. Let’s set $y = f \circ g(x)$:

$y = 2x + 2$

To find the inverse, solve for $x$ in terms of $y$:

$y - 2 = 2x$

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

So the inverse function is:

$(f \circ g)^{-1}(y) = \frac{y - 2}{2}$

Step 3: Evaluate $(f \circ g)^{-1}(6)$

Now, substitute $y = 6$ into the inverse function:

$(f \circ g)^{-1}(6) = \frac{6 - 2}{2} = \frac{4}{2} = 2$

Thus, the value of $(f \circ g)^{-1}(6)$ is $\boxed{2}$.

Let me know if you would like any further details or have any questions!

Here are 5 related questions:

1. How do you find the inverse of a function in general?
2. What is the difference between a function and its inverse?
3. Can you explain the process of finding the composite function $f \circ g$?
4. How does the order of functions affect the composite $f \circ g$ compared to $g \circ f$?
5. Why do we solve for $x$ when finding an inverse function?

Tip: When finding the inverse of a function, always swap the roles of $x$ and $y$, and then solve for $y$.