Question

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

• A. R
• B. $[1,\infty )$
• C. $[2,\infty )$
• D. $[5,\infty )$

Answer 1

Answer: (B)

Given $f:[2,\infty )\to B$ defined as f (x) = $x^2-4x+5$ for all x = 2, 3, ....where f (x) is bijection.
Since, f(x) is bijection (i.e. 1 - 1 and onto)
$\therefore$ Co-domain of f (x) = Range of f (x)
$\Rightarrow$ Range of f (x) = B
Now, Let $y=x^2-4x+5$
$\Rightarrow x^2=y+4x-5$
$\Rightarrow x^2-4x+5-y=0$
Since, this is a quadratic equation with constant term = 5 - y
$\therefore b^2-4ac\ge 0$
i.e. $(4{)}^2-4(1)(5-y)\ge 0$
$\Rightarrow 16-20+4y\ge 0$
$\Rightarrow y\ge 1$
$\Rightarrow y\in [1,\infty )$
Hence, $B=[1,\infty ).$