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Determine the domain of the following function t𝑓(𝑥) = 𝑥 – 3 / 𝑥 2 + 6𝑥 − 16

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Answer to a math question Determine the domain of the following function t𝑓(𝑥) = 𝑥 – 3 / 𝑥 2 + 6𝑥 − 16

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Rasheed
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Para determinar el dominio de la función \( t𝑓(𝑥) = \frac{𝑥 – 3}{𝑥^2 + 6𝑥 − 16} \), necesitamos encontrar el conjunto de todos los valores posibles de \( x \) que hará que la función esté definida. Esto significa que necesitamos identificar cualquier valor de \( x \) que causaría que el denominador sea cero, ya que la división por cero no está definida. El denominador de la función es una expresión cuadrática \( 𝑥^2 + 6𝑥 − 16 \). Para encontrar los valores de \( x \) que hacen que el denominador sea cero, podemos factorizar la cuadrática: 𝑥^2 + 6𝑥 − 16 = (𝑥 + 8)(𝑥 - 2) Igualar cada factor a cero nos da los valores de \( x \) que no están permitidos: 𝑥 + 8 = 0 \quad \Rightarrow \quad 𝑥 = -8 𝑥 - 2 = 0 \quad \Rightarrow \quad 𝑥 = 2 Por lo tanto, el dominio de la función \( t𝑓(𝑥) \) son todos los números reales excepto \( x = -8 \) y \( x = 2 \). En notación de intervalo, el dominio es: (-\infty, -8) \taza (-8, 2) \taza (2, \infty) Esto significa que la función \( t𝑓(𝑥) \) está definida para todos los números reales excepto \( x = -8 \) y \( x = 2 \).

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