The image contains a math problem in Greek regarding two functions $f(x)$ and $g(x)$ with the following tasks:

Given functions:

• $f(x) = \sqrt{x + 1}$
• $g(x) = 2 - x$

You are asked to:

1. Define the composition function $f \circ g$.
2. Prove that $f$ is invertible and find $f^{-1}$.
3. Determine the type of monotonicity (increasing or decreasing) of the function $f \circ g$.

Let’s work through each part:

i. Define the composition function $f \circ g$:

The composition $(f \circ g)(x)$ means applying $g(x)$ first and then applying $f$ to the result.

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

Now, substitute $g(x) = 2 - x$ into $f(x) = \sqrt{x + 1}$:

$f(g(x)) = \sqrt{(2 - x) + 1} = \sqrt{3 - x}$

So the composition is:

$(f \circ g)(x) = \sqrt{3 - x}$

ii. Prove that $f(x) = \sqrt{x + 1}$ is invertible and find $f^{-1}(x)$:

To prove that $f$ is invertible, we need to show that it is one-to-one (injective) and onto (surjective).

1. Injectivity: $f(x) = \sqrt{x + 1}$ is an increasing function because its derivative is positive. The derivative of $f(x)$ is:

   $f'(x) = \frac{1}{2\sqrt{x+1}} > 0 \text{ for all } x \geq -1$

   Since the derivative is always positive, $f(x)$ is strictly increasing, which implies it is one-to-one.

2. Finding the inverse: To find the inverse of $f(x) = \sqrt{x + 1}$, solve for $x$ in terms of $y$:

   $y = \sqrt{x + 1} \quad \Rightarrow \quad y^2 = x + 1 \quad \Rightarrow \quad x = y^2 - 1$

   Therefore, the inverse function is:

   $f^{-1}(x) = x^2 - 1$

iii. Determine the type of monotonicity of $f \circ g$:

To find the monotonicity of $(f \circ g)(x) = \sqrt{3 - x}$, we need to compute its derivative:

$(f \circ g)'(x) = \frac{d}{dx} \sqrt{3 - x} = \frac{-1}{2\sqrt{3 - x}}$

Since the derivative is negative for all $x$, the function $f \circ g$ is strictly decreasing.

Summary of Results:

1. $(f \circ g)(x) = \sqrt{3 - x}$
2. $f^{-1}(x) = x^2 - 1$
3. $(f \circ g)(x)$ is strictly decreasing.

Let me know if you would like further details or explanations.

Here are 5 related questions for further practice:

1. Find the domain of $f(x)$ and $g(x)$ separately.
2. What is the domain of the composition $f \circ g$?
3. Calculate the range of $f(x)$ and $f \circ g(x)$.
4. Verify the inverse $f^{-1}(x)$ by composing $f(f^{-1}(x))$ and $f^{-1}(f(x))$.
5. Analyze the monotonicity of $g(x) = 2 - x$ and compare it with that of $f(x)$.

Tip: Always check the domain restrictions when working with square roots or inverse functions.