Question

The number of linear functions which map $[-1,1]$ to $[0,2]$ are.

• A. One
• B. Two
• C. Four
• D. Three

Answer 1

The correct option is B Two

Let $f\left(x\right)=ax+b$, which is linear

Since $f\left(x\right)$is linear,it should be either increasing ordecreasing.

case(1); Let $f\left(x\right)$be increasing.

$\Rightarrow$ $F^{\prime }\left(x\right)\ge 0$

$\Rightarrow$ $a\ge 0$

$\Rightarrow$ $f(-1)=0andf\left(1\right)=2$(science $f$ is onto ,increasing in $[-1,1],$\to [0,2]$

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

solving these two equation ,we get

$a=b=1$

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

case (2); Let $f(x) be decreasing

$\Rightarrow$ $f\left(x\right)\le 0$

$\Rightarrow$ $f(-1)=2andf\left(1\right)=0$(science $f$ is onto ,decreasing in $[-1,1],$\to [0,2]$

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

solving these two equation ,we get

$a=-1,b=1$

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

the two linear equation are

$x+1and-x+1$

so answer is $2$.