Question

The function which map $[-1,1]$ to $[0,2]$ are

• A. One linear function
• B. Two linear function
• C. Circular function
• D. None of these

Answer 1

The correct option is B Two linear function

Let $f\left(x\right)=ax+b$ be a function that maps $[-1,1]$ to $[0,2]$ if $f\left(x\right)$ is increasing

$f^1\left(x\right)\ge 0$

$\Rightarrow a\ge 0$

Now, $f(-1)=0$ & $f\left(1\right)=2$

$\Rightarrow -a+b=0$ & $a+b=2$

$\Rightarrow a=b=1\Rightarrow f\left(x\right)=x+1$

if $f\left(x\right)$ is decreasing

$f^1\left(x\right)<0$

$\Rightarrow a\le 0$

$\therefore$ $f(-1)=2$ & $f\left(1\right)=0$

$\because$ $-a+b=2$ & $a+b=0$

$\therefore$ $a=-1$ & $b=1$

$\therefore$ $f\left(x\right)=-x+1$

Hence, $f\left(x\right)$ is two linear functions

i.e $f\left(x\right)=-x+1$ & $f\left(x\right)=1+x$ [B]