Question

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) = 2^x.

c) h: R → R given by h(x) = 5x - 1.

Answer 1

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