The distinct linear functions which map [-1, 1] onto [0, 2] are `f(x)=x+1,\ \ g(x)=-x+1` (b) `f(x)=x-1,\ \ g(x)=x+1` (c) `f(x)=-x-1,\ \ g(x)=x-1` (d) none of these

Options

A. `f(x)=x+1,g(x)=-x+1`
B. `f(x)=x-1,g(x)=x+1`
C. `f(x)=-x-1,g(x)=x+1`
D. None of these

Answer

A

Explanation

Let f(x)=ax+b be the required linear function. Then, f(x) is either strictly increasing or strictly decreasing
`therefore f'(x) gt 0 or, f'(x)lt0"for all " x in[-1,1]`
`Rightarrowagt 0 or, alt 0`
CASE-I When `a gt 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`
`Rightarrow -a+b=0 and a+b=2 Rightarrow a=b=1`
`therefore f(x)=x+1`
Case-II When `a lt 0`
In this case, f(x) is strictly decreasing and maps `[-1,1]"to" [0,2]`
Therefore, f(-1)=2 f(1)=0
`Rightarrow -a+b=2 and a+b=0 Rightarrow a=-1, b=1`
`therefore f(x)=-x+1`