Choose the correct answer from the given four options. Let f : [2… - Vidyadip

Question

Choose the correct answer from the given four options.
Let f : [2, ∞) → R be the function defined by f(x) = x² – 4x + 5, then the range of f is:

$\text{R}$
$[1,\infty)$
$[4,\infty)$
$[5,\infty)$

✓

Answer

$[1,\infty)$

Solution:

Given that, $\text{f}(\text{x})=\text{x}^2-4\text{x}+5,$

Let $\text{y}=\text{x}^2-4\text{x}+5$

$\Rightarrow\ \text{y}=\text{x}^2-4\text{x}+4+1$

$=(\text{x}-2)^2+1$

$\Rightarrow\ (\text{x}-2)^2=\text{y}-1$

$\Rightarrow\ \text{x}-2=\sqrt{\text{y}-1}$

$\Rightarrow\ \text{x}=2+\sqrt{\text{y}-1}$

$\therefore\ \text{y}-1\geq0,\ \text{y}\geq1$

Range $=[1,\infty)$

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