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 $J$.

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.