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Let's analyze the exercise in the image and go through each part in detail. Here is a breakdown of the problem:

We have a function \( g \) defined on the interval \( I = [1, +\infty[ \) with:
\[
g(x) = 2 - \sqrt{x^2 - 1}.
\]

The tasks are as follows:

### 1. **Calculate** \( \lim_{x \to +\infty} g(x) \).

For this, we need to evaluate the behavior of \( g(x) \) as \( x \) approaches infinity.

### 2. **Show that the function** \( g \) **is continuous on** \( I \).

To do this, we will need to verify that \( g(x) \) is defined and continuous on the interval \( [1, +\infty[ \), focusing on any potential points of discontinuity.

### 3. **Calculate** \( \lim_{x \to 1^+} \frac{g(x) - g(1)}{x - 1} \) **and interpret the result geometrically**.

This expression is essentially the definition of the derivative of \( g \) at \( x = 1 \). Calculating this limit will give insight into the behavior of \( g(x) \) near \( x = 1 \).

### 4. **Show that** \( \forall x \in ]1, +\infty[ \), \( g'(x) = \frac{-x}{\sqrt{x^2 - 1}} \). **Deduce the monotonicity of** \( g \) **on** \( I \).

This requires differentiating \( g(x) \) and analyzing the sign of \( g'(x) \) to determine if \( g(x) \) is increasing or decreasing over \( I \).

### 5. **Draw the variation table of** \( g \).

Using the results from the derivative and monotonicity, create a table that shows the behavior of \( g(x) \) over \( I \).

### 6. **Show that** \( g \) **admits an inverse function** \( g^{-1} \) **defined on an interval** \( J \) **to be determined**.

Here, we need to verify if \( g \) is bijective (one-to-one) on \( I \), allowing for the existence of an inverse function on a specific interval \( J \).

### 7. **Determine** \( g^{-1}(y) \) **for all** \( y \in J \).

Finally, we’ll find an explicit expression for the inverse function \( g^{-1}(y) \) on the interval \( J \).

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Would you like me to go through each part step-by-step with detailed solutions, or do you have specific questions about one of the tasks?

### Relative Questions
1. How do we calculate limits involving square roots as \( x \to +\infty \)?
2. What conditions must be met to prove a function is continuous on an interval?
3. How do we interpret the limit of a difference quotient geometrically?
4. What are the criteria for determining if a function is monotonic?
5. What steps are involved in finding an inverse function and determining its domain?

### Tip
When calculating limits with square roots, consider rationalizing by multiplying the numerator and denominator by the conjugate, if needed, to simplify the expression.

On considère la fonction g définie sur I = [1, +∞[ par: g(x) = 2 - √(x^2 - 1).

Explore a detailed solution for calculating limits, proving continuity, finding derivatives, analyzing monotonicity, and determining the inverse function for g(x) = 2 - √(x^2 - 1). This problem includes important calculus concepts such as limits, continuity, and inverse functions, suitable for university-level calculus students.

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