Question

find the domain of the real valued function f(x)=1/log(2-x)​

Answer 1

We have to find domain of the given function :-

$\rm \large f(x) = \dfrac{1}{log (2-x)}$

We know that denominator cannot be zero.

$\rm \longrightarrow log (2-x) \ne0 \\ \\ \rm \longrightarrow (2-x) \ne {e}^{0} \\ \\ \rm \longrightarrow 2-x \ne 1\\ \\ \rm \longrightarrow 2 - 1 \ne x\\ \\ \rm \longrightarrow 1 \ne x\\ \\ \rm \longrightarrow x \ne 1 \: \: \: \: ...(1)$

Also,

$\large \rm \longrightarrow 2 - x > 0 \\ \\ \large \rm \longrightarrow 2 > x \: \: ...(2)$

From equation (1) and (2) :-

$\rm D_f = x \in (- \infty, 2) - \{1 \} \\ or \\ \rm D_f = x \in (- \infty, 1) \cup (1,2) \\ \\ \rm \: where \: D_f \: is \: domain \: of \: the \: function$

Answer 2

f(x) = 1/(log(2 - x)) 2 - x > 0 and 2 - x ≠ 1 x < 2 and x ≠ 1 ∴ x ∈ (–∞, 1) ∪ (1, 2)Read more on Sarthaks.com - https://www.sarthaks.com/533036/find-the-domain-of-the-real-valued-function-f-x-1-log-2-x