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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions1 Mark
MCQ
A function $f: R \rightarrow R$ defined as $f(x)=x^2-4 x+5$ is:
  • A
    injective but not surjective
  • B
    surjective but not injective
  • C
    both injective and surjective
  • ✓
    neither injective nor surjective
✓

Answer

Correct option: D.
neither injective nor surjective
Given, $f(x)=x^2-4 x+5$
Here $f(0)=f(4)=5$
Hence, $f(x)$ is not one$-$one.
To check whether the function is onto or not,
we have to find range of function.
$\text { Let } y=x^2-4 x+5$
$\Rightarrow x^2-4 x+5-y=0$
$\therefore D=(4)^2-4(1)(5-y) \geq 0 \forall x \in R$
$\Rightarrow 16-20+4 y \geq 0$
$\Rightarrow 4 y-4 \geq 0$
$\Rightarrow 4(y-1) \geq 0$
$\Rightarrow y \geq 1$
Hence, range $=(1, \infty)$
Here, Co$-$domain $\neq$ Range
So, $f(x)$ is not onto.

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