I see that you've uploaded images of mathematical exercises, likely involving limits, derivatives, function properties, and curve sketching. I'll go over a detailed solution for each of these exercises to help you understand how to approach them.

Here’s a breakdown of each exercise from what I see in the first image:

Exercise 1: Limits

1. Calculate: $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$.
2. Interpret: Deduce what the behavior of the function $f$ indicates as $x \to +\infty$ and $x \to -\infty$.

For this, I would calculate the limits as $x$ approaches positive and negative infinity for $f(x)$, if the function expression is provided. Then, based on the limits, interpret the end behavior of the function.

Exercise 2: Derivative and Interpretation

• Study the derivative of function $f$ and interpret the result.

Here, you would differentiate $f(x)$ with respect to $x$ and interpret what the sign of $f'(x)$ tells us about the monotonicity (increasing or decreasing behavior) of $f$.

Exercise 3: Function Properties

1. Show that for all $x \in [0; +\infty[$, the function $f(x)$ meets specific conditions.
2. Study the sign of $f(x)$ and construct the variation table for $f$.

This step would involve analyzing the sign changes of $f(x)$ over its domain and creating a table to visualize where $f(x)$ is increasing or decreasing.

Exercise 4: Concavity and Inflection Points

• Study the concavity of the curve and identify any inflection points.

To analyze concavity, take the second derivative $f''(x)$ and evaluate where it changes sign, indicating inflection points where concavity changes.

Exercise 5: Solve and Interpret

• Solve the equation in $\mathbb{R}^+$ and interpret the solution graphically.

This involves solving $f(x) = x$ within the specified domain $\mathbb{R}^+$ and interpreting what this means for the graph of $f$.

Exercise 6: Curve Sketching of $C_f$

• Sketch the curve $C_f$ in a given coordinate system.

This requires drawing the graph of $f(x)$ based on the analysis above, including any intercepts, asymptotes, maxima, minima, and concavity.

The second image seems to involve similar tasks with another function $g$, particularly focusing on:

1. Restriction of function $g$, interval determination, and verifying whether $g$ admits an inverse.
2. Calculating $g(g^{-1}(9))$ and confirming properties of $g$.
3. Sketching the curve of $g$.

Would you like detailed, step-by-step solutions for each of these exercises?

Follow-up Questions

1. Would you like help with specific calculations, such as limits or derivatives?
2. Do you need assistance in creating the variation table for $f(x)$?
3. Are you looking for a guide on sketching the curves based on the properties found?
4. Do you want detailed steps on how to find concavity and inflection points?
5. Would you like an explanation of how to interpret the graphical behavior after solving equations?

Tip

When analyzing functions for sketching, it’s essential to look at critical points, asymptotes, and end behaviors systematically for an accurate graph.