If a function f: (2, ) B defined by f(x)=x^2-4 x+5 is a bijection… - Vidyadip

MCQ

If a function $f:\ (2, \infty) \rightarrow B$ defined by $f(x)=x^2-4 x+5$ is a bijection, then $B=$

A
$\text{R}$
✓
$(1,\infty)$
C
$(4,\infty)$
D
$(5,\infty)$

✓

Answer

Correct option: B.

$(1,\infty)$

Since $f$ is a bijection$, co-$domain of $f=$ range of $f$
$\Rightarrow B=$ range of $f$
Given: $f(x)=x^2-4 x+5$
Let $f(x)=y$
$\Rightarrow y=x^2-4 x+5$
$\Rightarrow x^2-4 x+(5-y)=0$
$\because$ Discrimant, $\text{D}=\text{b}^2-4\text{ac}\geq0,$
$(-4)^2-4\times1\times(5-\text{y})\geq0$
$\Rightarrow\ 16-20+4\text{y}\geq0$
$\Rightarrow\ 4\text{y}\geq4$
$\Rightarrow\ \text{y}\geq1$
$\Rightarrow\ \text{y}\in[1,\infty)$
$\Rightarrow$ Range of $\text{f}=[1,\infty)$
$\Rightarrow\ \text{B}=[1,\infty)$

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