Question

find the domain of fx=sin^-1log6(3x)​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm :\longmapsto\:y = {sin}^{ - 1}( log_{6}(3x))$

We know,

$\rm :\longmapsto\: {sin}^{ - 1}x \: is \: defined \: if$

$\rm :\longmapsto\: - 1 \leqslant x \leqslant 1$

So it implies,

$\rm :\longmapsto\:y = {sin}^{ - 1}( log_{6}(3x)) \: is \: defined \: if \:$

$\rm :\longmapsto\: - 1 \leqslant log_{6}(3x) \leqslant 1$

$\rm :\longmapsto\: {6}^{ - 1} \leqslant 3x \leqslant {6}^{1}$

$\: \: \: \: \: \: \green{\bigg \{ \because \: \tt \: log_{x}(y) = z \implies \: y = {x}^{z} \bigg \}}$

$\rm :\longmapsto\: \dfrac{1}{6} \leqslant 3x \leqslant {6}^{1}$

$\rm :\longmapsto\: \dfrac{1}{18} \leqslant x \leqslant 2$

$\bf\implies \:x \: \in \: \bigg[\dfrac{1}{18}, \: 2 \bigg]$

Additional Information :-

Domain :- Let f(x) be a function, then set of those values of x where f(x) is well defined is called domain.

Range :-

To find the range of f(x)

Step : - 1. Let y = f(x)

Step :- 2. Express x in terms of y, say x = g(y).

Step :- 3. Find the domain of g(y).

Step :- 4. This will be the range of f(x).