"Inversas" que no conmutan.

A continuación se demuestra, aplicando diferentes propiedades aritméticas (y algunos truquitos) que si [math]9=4+5[/math], entonces [math]3=2[/math].[br][br]Lógicamente, hay un error en alguno de los pasos, ¿podrás averiguarlo?
¿En qué paso se comete el error?[br]¿Qué propiedad están aplicando de manera incorrecta?
Veamos desde el punto de vista de la composición de funciones y las funciones inversas, porqué sucede esto.[br][br]Consideremos la función [math]f\left(x\right)=x^2[/math]. como todos sabemos, la operación contraria a elevar al cuadrado es la raíz cuadrada. Entonces podríamos suponer que la función inversa de [math]f\left(x\right)[/math] es [math]f^{-1}\left(x\right)=\sqrt{x}[/math].[br][br]Según la definición de la función inversa se tiene que cumplir que [math]f\circ f^{-1}\left(x\right)=f^{-1}\circ f\left(x\right)=id\left(x\right)=x[/math].[br][br]Observa las graficas de las dos funciones, así como de las composiciones entre ellas manipulando el siguiente Applet:
¿Cómo es la gráfica de [math]f\left(x\right)[/math]?[br]¿Cuál es su dominio?
¿Cómo es la gráfica de [math]f^{-1}\left(x\right)[/math]?[br]¿Cuán es su dominio?
¿Se cumple esta vez que [math]f\circ f^{-1}\left(x\right)=f^{-1}\circ f\left(x\right)[/math] ?
Como conclusión, diremos que la función [math]f\left(x\right)=x^2[/math] no tiene inversa, en general.[br][br]Igualmente, en general no podemos afirmar que [br][img]data:image/png;base64,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[/img][br]ya que esto solo es cierto si [math]x\ge0[/math].[br][br]La simplificación de raíces con cuadrados podríamos expresarla con dos propiedades:[br][br][img]data:image/png;base64,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[/img][br][img]data:image/png;base64,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[/img]
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