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  4. Let f:[0, ∞) arrow[0, ∞) be a function defined by f(x)=x2+(k2-3 k+2) x+k2-k. If f(x) is both injective and surjective, then the number of integers in the range of k is

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Q. Let $f:[0, \infty) \rightarrow[0, \infty)$ be a function defined by $f(x)=x^2+\left(k^2-3 k+2\right) x+k^2-k$. If $f(x)$ is both injective and surjective, then the number of integers in the range of $k$ is

Relations and Functions - Part 2

Solution:

image
For $f ( x )$ to be injective and surjective
$f (0)=0 \text { and } \frac{- b }{2 a }=0 $
$\therefore k ^2-3 k +2=0 \text { and } k ^2- k =0 \Rightarrow k =1$
OR
image $D >0 \& f (0)=0 \Rightarrow k =0$