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Archangel Macsika

Which of the following functions are injective? Which are surjective? a) f: Z → Z given by f(x) = x2 + 1. b) g: N → N given by g(x) = 2x. c) h: R → R given by h(x) = 5x - 1.

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a) f:\ \mathbb{Z}\rightarrow \mathbb{Z} ,\quad f(x)=x^2+1

1) f(-1)=f(1)=2

f is not injective

2) Since f(x)=x^2+1\geq 1 , there doesn’t exists any x\in\mathbb{Z} such that f(x)=0

f is not surjective


b) g:\ \mathbb{N}\rightarrow \mathbb{N} , g(x)=2^x

1) If g(x)=g(y) , 2^x=2^y , then x=y

f is injective

2) g(x)=2^x=1 only if x=0 , but 0\not \in \mathbb{N}

f is not surjective


c) h:\ \mathbb{R}\rightarrow \mathbb{R} , h(x)=5x-1

1) If h(x)=h(y) , 5x-1=5y-1 , 5x=5y , then x=y

f is injective

2) For all y\in\mathbb{R} there exists x=\frac{y+1}{5} such that f(x)=5x-1=y+1-1=y

f is surjective


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Question ID: mtid-5-stid-8-sqid-1231-qpid-969