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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions1 Mark
MCQ
The distinct linear functions that map $[-1, 1]$ onto $[0, 2]$ are:
  • A
    $f(x) = x + 1, g(x) = -x + 1$
  • B
    $f(x) = x - 1, g(x) = x + 1$
  • ✓
    $f(x) = -x - 1, g(x) = x - 1$
  • D
    None of these.
✓

Answer

Correct option: C.
$f(x) = -x - 1, g(x) = x - 1$
Since $f$ is invertible, range of $f = co-$domain of $f = x$
So, we need to find the range of $f$ to find $X.$
For finding the range, let $f(x) = y$
$\Rightarrow 4 x - x ^2= y$
$\Rightarrow x ^2-4 x =- y$
$\Rightarrow x ^2-4 x +4=4- y$
$\Rightarrow( x -2)^2=4- y$
$\Rightarrow\ \text{x}-2=\pm4-\text{y}$
$\Rightarrow\ \text{x}=2\pm4-\text{y}$
This is defined only when $4-\text{y}\geq0$
$\Rightarrow\ \text{y}\leq4,$
$X =$ Range of $f =(-\infty,4]$

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