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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 \( y \).

Diketahui fungsi f dan g yang ditentukan oleh f(x) = x + 3 dan g(x) = 2x - 1. Nilai dari (f ∘ g)⁻¹(6) adalah ....

Learn how to calculate the inverse of a composite function with f(x) = x + 3 and g(x) = 2x - 1, and find the value of (f ∘ g)⁻¹(6). This problem covers essential algebra concepts, such as function composition and inverses, suitable for students in Grades 9-12.

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