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The image contains some functional compositions and inverse functions, particularly focusing on combinations of functions \( f \), \( g \), and their inverses. Let me transcribe and explain the key parts:

1. \((f \circ g)(x) = f(g(x))\)
   - This is the standard notation for the composition of two functions. It means applying the function \( g \) to \( x \), then applying \( f \) to the result.
   
2. \((g \circ f)(x) = g(f(x))\)
   - Similarly, this is the composition of \( g \) and \( f \), where \( f(x) \) is computed first and then the result is passed to \( g \).

3. \((f \circ g)^{-1}(x) = g^{-1}(f^{-1}(x))\)
   - This equation shows the inverse of the composition of functions. It follows the rule: to invert a composition of functions, you reverse the order and take the inverse of each function.

4. \( f(g(x)) = f \circ g \)
   - This again represents the composition of \( f \) and \( g \), expressed in a slightly different form.

5. \( F(2-3) \) and \( F(3) \)
   - These expressions likely represent specific values being plugged into a function \( F \), although it’s not fully clear from the context what \( F \) represents.

6. \( (f \circ g)(c) \)
   - This indicates the composition of \( f \) and \( g \) applied to a constant \( c \).

Would you like further clarification on one of these parts or a detailed step-by-step explanation of any specific equation? 

Here are five related questions that may help deepen your understanding:

1. How do we find the inverse of a function, and what does it mean geometrically?
2. Why does the order of operations reverse when taking the inverse of a composition?
3. Can every function have an inverse? If not, what are the conditions for a function to be invertible?
4. What is the significance of function composition in real-world applications?
5. How do we solve equations involving compositions of functions?

**Tip:** When dealing with compositions and inverses, always pay attention to the order of operations as it plays a critical role in both solving and simplifying equations.

The user submitted an image containing function compositions and inverse functions involving the functions f and g.

Explore function composition and inverse functions through examples involving the functions f and g. Learn how to solve expressions like (f ∘ g)(x), (g ∘ f)(x), and their inverses with step-by-step explanations. Suitable for students in Grades 11-13.

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