Question

If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =

Options

• R

• [1, ∞)

• [4, ∞)

• [5, ∞)

Answer 1

Since f is a bijection, co-domain of f = range of f

⇒ B = range of f

\[\text{Given}: f\left( x \right) = x^2 - 4x + 5\] 
\[\text{Let}f\left( x \right) = y\] 
\[ \Rightarrow y = x^2 - 4x + 5\] 
\[ \Rightarrow x^2 - 4x + \left( 5 - y \right) = 0\] 
\[ \because \text{Discrimant}, D = b^2 - 4ac \geq 0, \] 
\[ \left( - 4 \right)^2 - 4 \times 1 \times \left( 5 - y \right) \geq 0\] 
\[ \Rightarrow 16 - 20 + 4y \geq 0\] 
\[ \Rightarrow 4y \geq 4\] 
\[ \Rightarrow y \geq 1\] 
\[ \Rightarrow y \in [1, \infty )\] 
\[ \Rightarrow \text{Range of f} = [1, \infty )\] 
\[ \Rightarrow B = [1, \infty )\]

So, the answer is (b).