The number of linear functions which map [ $-1,1$ ] onto $[0,2]$ is

Question

Professor PC Thomas Classes · Answer

A linear function from $[-1,1]$ to $[0,2]$ is of the form $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}, \mathrm{a} 
eq 0$. If $\mathrm{a}>0$ it is increasing and if $\mathrm{a}<0$ it is decreasing. As f is onto there are 2 possibilities. \ i) $f(-1)=0 \quad f(1)=2$ \ ii) $f(-1)=2 \quad f(1)=0$ \ i) $\Rightarrow-a+b=0 a+b=2$ $ \Rightarrow \mathrm{a}=\mathrm{b}=1 $ \ ii) $\Rightarrow-\mathrm{a}+\mathrm{b}=2 \mathrm{a}+\mathrm{b}=0 \Rightarrow \mathrm{a}=-1 \mathrm{~b}=1$. So there are 2 linear functions.