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Mathematik-Online problems:

Problem 10: Injective, Surjective, Bijective

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Which of the following maps are injective, surjective, respectively bijective? In each case give the codomain $ W(f)$.
a) $ f: \mathbb{R} \longrightarrow \mathbb{R}, \ x \longmapsto \vert x\vert-1$  b) $ f: \{x\in\mathbb{R}\mid x\geq 0\} \longrightarrow \mathbb{R}, \ x \longmapsto {\displaystyle{\frac{1}{1+x^2}}}$
c) $ f: \mathbb{R} \longrightarrow \mathbb{R}, \ x \longmapsto x\sqrt{1+x^2}$  d) $ f: \mathbb{R} \longrightarrow \mathbb{R}, \ x \longmapsto \min(x,x^2)$

For which $ a\in\mathbb{R}$ the following maps are injective?
e) $ f: \mathbb{R} \longrightarrow \mathbb{R}, \ x \longmapsto x^3+ax$  f) $ f: \mathbb{R}\setminus\{2\} \longrightarrow \mathbb{R}, \ x \longmapsto {\displaystyle{\frac{2x^2-3x+a}{x-2}}}$
g) $ f: [0,a] \longrightarrow \mathbb{R}, \ x \longmapsto x^2-5x+7$     

(Authors: Kimmerle/Roggenkamp/Höllig/Höfert)

see also:

[Solutions]
  automatisch erstellt am 18.  1. 2017