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1. Construis les courbes représentatives respec- tives \( (\mathscr{C}) \) et \( \left(\mathscr{C}_{g}\right) \) des fonctions \( f \) et \( g \) définies sur \( \mathbb{R}^{*} \) par \( : g(x)=x-2 \) et \( h(x)=\frac{1}{x} \). 2. Compare graphiquement les courbes \( \left(\mathscr{C}_{\mathcal{C}}\right) \) et \( \left(\mathscr{C}_{\mathcal{G}}\right) \). 3. Vérifie les résultats algébriquement.

Ask by Wade Wood. in Mali
Mar 09,2025

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1. **Construction des courbes représentatives** - **Fonction \( g(x) = x - 2 \)**: - Droite avec pente 1 et coupant l'axe des ordonnées en \( (0, -2) \). - Point supplémentaire : \( (2, 0) \). - **Fonction \( h(x) = \frac{1}{x} \)**: - Hyperbole avec asymptotes verticale en \( x = 0 \) et horizontale en \( y = 0 \). - Deux branches dans les demi-plans \( x > 0 \) et \( y > 0 \), et \( x < 0 \) et \( y < 0 \). 2. **Comparaison graphique** - Les deux courbes se coupent en deux points. - Ces points d'intersection correspondent aux solutions de l'équation \( x - 2 = \frac{1}{x} \). 3. **Vérification algébrique** - Résolution de \( x - 2 = \frac{1}{x} \) : - Multiplie par \( x \) : \( x^2 - 2x = 1 \). - Équation quadratique : \( x^2 - 2x - 1 = 0 \). - Discriminant \( \Delta = 8 \). - Solutions : \( x = 1 \pm \sqrt{2} \). - Points d'intersection : \( \left(1 + \sqrt{2}, \sqrt{2} - 1\right) \) et \( \left(1 - \sqrt{2}, -1 - \sqrt{2}\right) \). Ces points confirment que les courbes \( \mathscr{C} \) et \( \mathscr{C}_{g} \) se coupent bien en deux endroits.

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Bonus Knowledge

Pour construire les courbes représentatives des fonctions \( g(x) = x - 2 \) et \( h(x) = \frac{1}{x} \), on peut les tracer sur un même graphique. La fonction \( g \) est une droite qui coupe l'axe des ordonnées au point \( (0, -2) \) avec une pente de 1, tandis que \( h \) est une hyperbole. Elle n'est pas définie pour \( x = 0 \), montrant deux branches : une dans le quadrant I et une dans le quadrant III. Graphiquement, on observe que \( \mathscr{C}_{g} \) est linéaire et continues, tandis que \( \mathscr{C}_{h} \) présente une asymptote vertical à \( x=0 \) et une horizontale à \( y=0 \). Au fur et à mesure que \( x \) tend vers 0, \( h(x) \) approche l'infini, surtout lorsque \( x \) devient très petit. Algébriquement, on peut comparer les valeurs des fonctions pour un ensemble de \( x \) afin de montrer comment elles se comportent les unes par rapport aux autres sur leur domaine respectif.

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