Browse Questions

# Find the linear functions that maps $[-1,1]$ onto $[0,2]$.

$\begin{array}{1 1} x-1,x+1 \\ x-1,-x-1 \\ x+1,-x+1 \\ x-1,-x+1 \end{array}$

Let $f(x)=ax+b,$ which is linear.
Since $f(x)$ is linear, it should be either increasing or decreasing.
Case (i); Let $f(x)$ be increasing.
$\Rightarrow\:f'(x)\geq 0$
$\Rightarrow\:a\geq 0$
$\Rightarrow\: f(-1)=0\:and\:f(1)=2$ (Since $f$ is onto,increasing in $[-1,1]\rightarrow\:[0,2]$).
$\Rightarrow\:-a+b=0\:\:and\:\:a+b=2$
Solving these two equations we get $a=b=1$
$\Rightarrow f(x)=x+1$
Case (ii): Let $f(x)$ be decreasing .
$\Rightarrow\:f'(x)\leq 0$
$\Rightarrow \:a\leq 0$
$\Rightarrow f(-1)=2\:\:and\:\:f(1)=0$ (Since $f$ is onto, decreasing in $[-1,1]\rightarrow [0,2])$.
$\Rightarrow -a+b=2\:\:and\:\:a+b=0$.
Solving these two equations we get
$a=-1,\:b=1$
$f(x)=-x+1$
The two linear equations are
$x+1\:\:and\:\:-x+1$