Question

If there are exactly two distinct linear functions which map's from $[-1,1]\text{to}[0,2].$ Then those functions are

• A. $x+1$
• B. $-x+1$
• C. $x-1$
• D. $-1-x$

Answer 1

Answer: (A), (B)

$-x+1$

Since, there are exactly two distinct linear functions which map's from $[-1,1]\text{to}[0,2].$
So, Range $=[0,2]$
Let the required linear function be $f\left(x\right)=ax+b$

For a linear function $f\left(x\right)=ax+b$, in the interval $[x_1,x_2]$,we know that
$\left(i\right)$ range is $\left[f\right(x_1),f(x_2\left)\right]$, if $a>0$
$\left(ii\right)$ range is $\left[f\right(x_2),f(x_1\left)\right]$, if $a<0$

Case $\text{I}:$ If $a>0,$
$f(-1)=0,f\left(1\right)=2$
$\Rightarrow -a+b=0,a+b=2$
$\Rightarrow a=1,b=1$

Case $\text{II}:$ If $a<0,$
$f(-1)=2,f\left(1\right)=0$
$-a+b=2,a+b=0$
$\Rightarrow a=-1,b=1$
$\therefore$ Required functions are $f\left(x\right)=x+1\text{(or)}-x+1$