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This exercise requires us to analyze the function \( f(x) = x - 2\sqrt{x + 1} \) over its domain \([-1, +\infty[\). Let's go through the different questions in the exercise step by step.

### Solution Outline

1. **Calculer \(\lim_{x \to +\infty} f(x)\):**
   - As \(x\) approaches \(+\infty\), analyze the behavior of \(f(x)\) to find the limit.

2. **Étudier la continuité de \(f\) sur l'intervalle \([-1, +\infty[\):**
   - Investigate the continuity of \(f(x)\) over the interval. Consider if there are any points of discontinuity.

3. **Calculations of \(f'(x)\) and the variation table:**
   - **(a)** Calculate the derivative \(f'(x)\) for \(x \in ]-1, +\infty[\).
   - **(b)** Use \(f'(x)\) to determine the intervals of increase and decrease of \(f(x)\), and draw the table of variations.

4. **Analyzing the restriction \(g\) of \(f\) over \(]0, +\infty[\):**
   - Define \(g(x) = f(x)\) restricted to \(]0, +\infty[\).
   - **(a)** Show that \(g\) admits an inverse function \(g^{-1}\) over a suitable interval \(J\).
   - **(b)** Verify the expression for \(g(x)\) given as \((\sqrt{x+1} - 1)^2 - 2\).
   - **(c)** Find the expression of \(g^{-1}(x)\).
   - **(d)** Determine the variation table for \(g^{-1}(x)\).

### Starting with Step-by-Step Solutions:

#### 1. Calculating \(\lim_{x \to +\infty} f(x)\)
For large values of \(x\), the term \(\sqrt{x + 1}\) behaves similarly to \(\sqrt{x}\), so we can simplify:
\[
f(x) = x - 2\sqrt{x + 1} \approx x - 2\sqrt{x} = x - 2x^{1/2}.
\]
Using limits, we can determine the behavior as \(x \to +\infty\).

Would you like to proceed with the detailed solutions step-by-step, starting from the limit, or would you prefer a specific part to be explained in detail?

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### Follow-up Questions

1. What is the effect of removing the square root in large values of \(x\) when calculating limits?
2. How does the continuity of a function relate to its differentiability on an interval?
3. Why is calculating the derivative essential for understanding the variation table of a function?
4. How can we check if a function is invertible over a given interval?
5. What does it mean for a function to have a reciprocal over an interval?

### Tip

For complex functions involving roots, try to analyze the dominant terms as \(x\) grows large, which can simplify calculating limits and understanding asymptotic behavior.

Given the function f(x) = x - 2√(x + 1) defined on the interval [-1, +∞[, solve the following:
1. Calculate lim x→+∞ f(x).
2. Study the continuity of f on the interval [-1, +∞[.
3. Calculate f'(x) for x in ]-1, +∞[ and determine the variation table of f.
4. Let g be the restriction of f on ]0, +∞[. Show that g has an inverse function g⁻¹ defined on an interval J to be determined, verify the expression of g(x), deduce the expression of g⁻¹(x), and create the variation table of g⁻¹.

Explore the properties of the function f(x) = x - 2√(x + 1) on the interval [-1, +∞[, including calculations of limits, continuity analysis, derivative calculations, and conditions for invertibility. Suitable for high school students in Grades 11-12.

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