Question

Which of the following are two distinct linear functions which map the interval $[-1,1]$ onto $[0,2]$

• A. $f\left(x\right)=1+x$ or $1-x$
• B. $f\left(x\right)=1+2x$ or $1-x$
• C. $f\left(x\right)=1+x$ or $1-2x$
• D. $f\left(x\right)=1+x$ or $2-x$

Answer 1

The correct option is A $f\left(x\right)=1+x$ or $1-x$

Out of two linear functions one will be increasing and other will be decreasing.

Let $f\left(x\right)=ax+b$

For increasing: $f(-1)=0=>b-a=0\Rightarrow b=a$ and $f\left(1\right)=2\Rightarrow a+b=2\Rightarrow a=b=1\therefore f\left(x\right)=1+x$

For decreasing: $f(-1)=2=>b-a=2$ and $f\left(1\right)=0\Rightarrow a+b=2\Rightarrow b=1,a=-1\therefore f\left(x\right)=1-x$