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Funciones: Inversa y Composición

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Funciones: Inversa y Composición

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Quiz sobre inversa y composición de funciones

Created by

Magalis Royero
Magalis Royero
Colombia

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Funciones: Inversa y Composición
 

Funciones: Inversa y ComposiciónOnline version

Quiz sobre inversa y composición de funciones

by Magalis Royero
1

Si f(x)=2x+3, ¿cuál es la inversa f^{-1}(x)?

2

¿Qué condición debe cumplir f para que exista f^{-1}?:

Escoge una o varias respuestas

3

Si g(x)=x^2, ¿g no tiene inversa en R?

4

Calcula (f∘g)(x) para f(x)=3x+1 y g(x)=x-2.

5

Si h(x)=√x y k(x)=x+4, ¿(h∘k)(x)?

6

Si f(x)=x−7 y f^{-1}(x)=x+7, ¿cuál es la imagen de 5 bajo f?

7

Propiedad de la composición: (f∘g)^{-1} = g^{-1}∘f^{-1} ¿Verdadero o falso?

8

Si f(x)=1/x y g(x)=2x, ¿(f∘g)(x)?

9

Si f(x)=3−x y h(x)=2x, ¿(h∘f)(x)?

10

Qué paso define que f y f^{-1} se “anulan” al componer:

Explicación

La inversa se obtiene despejando x: y=2x+3 => x=(y-3)/2.

Una inversa existe si f es biyectiva.

x^2 no es biyectiva sobre R porque no es inyectiva; solo en [0,∞) tiene inversa.

Sustituye g(x) en f: 3(x-2)+1.

Sustituye k(x) en h: √(x+4).

Si f^{-1}(5)=12, entonces f(12)=5; invierte para f(−2)→ no aplica; usar f(−2)=−9 correcta? Nota: pregunta requiere entender inversa: f(−2)=−9; para 5, si f(a)=5, a=f^{-1}(5)=12; entonces pregunta errónea; ajusto: calcular f(5)=−2.

Para funciones biyectivas, la inversa de la composición se invierte en orden.

Sustituyes g en f: 1/(2x).

Sustituye f en h: 2(3−x).

Propiedad de inversas: la composición en el orden correcto devuelve la identidad.

 
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