A function f: R R defined as f(x)=x^2-4 x+5 is: - Vidyadip

MCQ

A function $f: R \rightarrow R$ defined as $f(x)=x^2-4 x+5$ is:

A
injective but not surjective
B
surjective but not injective
C
both injective and surjective
D
neither injective nor surjective

✓

Answer

Given, $f(x)=x^2-4 x+5$Here $f(0)=f(4)=5$
Hence, $f(x)$ is not one-one.
To check whether the function is onto or not, we have to find range of function.
\[
\begin{array}{l}
\text { Let } y=x^2-4 x+5 \Rightarrow x^2-4 x+5-y=0 \\
\therefore \quad D=(4)^2-4(1)(5-y) \geq 0 \forall x \in R \\
\Rightarrow \quad 16-20+4 y \geq 0 \Rightarrow 4 y-4 \geq 0 \\
\Rightarrow \quad 4(y-1) \geq 0 \Rightarrow y \geq 1
\end{array}
\]
Hence, range $=(1, \infty)$
Here, Co-domain $\neq$ Range
So, $f(x)$ is not onto.

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