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Home » Let \( y = f(x) \), so \( f(y) = x \). Then: - Aurero

Let \( y = f(x) \), so \( f(y) = x \). Then: - Aurero

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 Lösung: Um \( f(x^2 - 2) \) zu finden, beginnen wir damit, die Struktur von \( f(x^2 + 2) = 3x^4 + 12x^2 + 7 \) zu analysieren.

📌 Erstens bemerken wir, dass \( 3x^4 + 12x^2 + 7 \) umgeschrieben werden kann, um eine Substitution widerzuspiegeln, die zur Form von \( f \) passt. Sei \( y = x^2 + 2 \). Dann ist \( x^2 = y - 2 \), und wir substituieren dies in das gegebene Polynom ein:

📌 f(y) = 3(y - 2)^2 + 12(y - 2) + 7

📌 Nun expandieren und vereinfachen wir diesen Ausdruck:

📌 f(y) &= 3(y^2 - 4y + 4) + 12y - 24 + 7 \\

📌 &= 3y^2 - 12y + 12 + 12y - 24 + 7 \\

📌 &= 3y^2 - 5

📌 Somit ist die Funktion \( f(y) = 3y^2 - 5 \). Nun sustituieren wir \( y = x^2 - 2 \), um \( f(x^2 - 2) \) zu finden:

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