The distinct linear functions which map [-1, 1] onto [0, 2] are

$f(x) = x+1, g(x) = -x+1$

The correct answer is Option (1) → $f(x) = x+1, g(x) = -x+1$

Let $f(x) = ax + b$ be the required linear function. Then, f(x) is either strictly increasing or strictly decreasing.

$∴f'(x) > 0$ or, $f'(x) <0$ for all $x∈[-1,1]$

$⇒a>0$ or, $a<0$

CASE I When $a > 0$

In this case $f(x) = ax + b$ is strictly increasing and maps [−1, 1] onto [0, 2]. Therefore,

$f(-1) = 0$ and $f(1) = 2$

$⇒-a+b=0$ and $a+b=2⇒ a = b = 1$

$∴f(x) = x + 1$

CASE II When $a <0$

In this case, f(x) is strictly decreasing and maps [-1, 1] to [0, 2]. Therefore, $f (-1) = 2$ and $f (1) = 0$

$⇒-a+b=2$ and $a+b=0⇒a=-1,b=1$

$∴f(x) = -x+1$

Hence, the distinct functions are $f(x) = x+1$ and $g(x) = −x+1$.