Answer to Question #185759 in Discrete Mathematics for zain ul abdeen

Question

Let f be a function from Z to R, such that f(x)=x/10, then f is

a) an increasing function   

b) a strictly increasing function

c) a decreasing function

d) an onto function

Answer 1

Let $f$ be a function from $\mathbb Z$ to $\mathbb R$, such that $f(x)=\frac{x}{10}$. If $x<y$, then $\frac{x}{10}<\frac{y}{10}$, and hence $f(x)<f(y)$. It follows that $f$ is a strictly increasing function.

The function $f$ is not an onto function. Indeed, for $y=\frac{1}{20}$ the equation $f(x)=\frac{1}{20},$ which is equivalent to $\frac{x}{10}=\frac{1}{20}$ and hence to $x=\frac{1}{2},$ has no solution in the set $\mathbb Z$ of integer numbers. Therefore, there is no integer number $x$ such that $f(x)=\frac{1}{20}.$