Question

If a function $f:[2,\infty )\to B\mathrm{defined}\mathrm{by}f\left(x\right)=x^2-4x+5$ is a bijection, then B =
(a) R
(b) [1, ∞)
(c) [4, ∞)
(d) [5, ∞)

Answer 1

Since f is a bijection, co-domain of f = range of f
$\Rightarrow$B = range of f
$$\begin{gathered} \text{Given}:f\left(x\right)=x^2-4x+5 \\ \mathrm{Let}f\left(x\right)=y \\ \Rightarrow y=x^2-4x+5 \\ \Rightarrow x^2-4x+\left(5-y\right)=0 \\ \because \mathrm{Discrimant},D=b^2-4ac\ge 0, \\ {\left(-4\right)}^2-4\times 1\times \left(5-y\right)\ge 0 \\ \Rightarrow 16-20+4y\ge 0 \\ \Rightarrow 4y\ge 4 \\ \Rightarrow y\ge 1 \\ \Rightarrow y\in [1,\infty ) \\ \Rightarrow \mathrm{Range}\mathrm{of}f=[1,\infty ) \\ \Rightarrow B=[1,\infty ) \end{gathered}$$
So, the answer is (b).