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Properties of Inverse Function

The set of va...

Question

The set of value(s) of $x$ for which $f (x) = e^{ln (- x^{2} + 5 x - 6)}$ and $g (x) = - x^{2} + 5 x - 6$ are identical functions is

[2, 3]

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(2, 3)

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(- \infty, 2) \cup (3, \infty)

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R - {2, 3}

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Solution

The correct option is B $(2, 3)$
Given : $f (x) = e^{ln (- x^{2} + 5 x - 6)} and g (x) = - x^{2} + 5 x - 6$
For domain of $f (x)$ :
$- x^{2} + 5 x - 6 > 0$
$\Rightarrow (2 - x) (x - 3) > 0 \Rightarrow x \in (2, 3)$
and domain of $g (x) = - x^{2} + 5 x - 6$ is $R .$
and $f (x) = e^{ln (- x^{2} + 5 x - 6)} = - x^{2} + 5 x - 6$
$\Rightarrow$ Both functions will be identical if $D_{f} = D_{g}, R_{f} = R_{g}$
So, $x \in (2, 3)$

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