Question

How do you find the domain and range of $f(x) = \arcsin (3 - 2x)$ ?

Answer 1

The domain of a function is the complete step of possible values of the independent variable. That is the domain is the set of all possible ‘x’ values which will make the function ‘work’ and will give the output of ‘y’ as a real number. The range of a function is the complete set of all possible resulting values of the dependent variable, after we have substituted the domain.

Complete step by step solution:
The domain of the expression is all real numbers except where the expression is undefined means There are no radicals or fractions involved. In this case, there is no real number that makes the expression undefined.
Given, $f(x) = \arcsin (3 - 2x)$
The inverse is also represented by arc, therefore the given function can be written as
$\Rightarrow f(x) = si{n^{ - 1}}(3 - 2x)$
Now, let's define the domain and range of $y = {\sin ^{ - 1}}x$.
The range of the function $y = {\sin ^{ - 1}}(x)$ is $[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} ]$.
The domain of the function $y = {\sin ^{ - 1}}(x)$ is $[ { - 1, + 1} ]$.
Here in this question we don’t have $y = {\sin ^{ - 1}}x$, we have $y = si{n^{ - 1}}(3 - 2x)$, so the domain will change
That means $- 1 \leqslant 3 - 2x \leqslant 1$
Now add -3 to the above inequality we have
$\Rightarrow - 1 - 3 \leqslant 3 - 2x - 3 \leqslant 1 - 3$
On simplifying we have
$\Rightarrow - 4 \leqslant - 2x \leqslant - 2$
Now we will multiply the above inequality by -1 and we have
$\Rightarrow 4 \geqslant 2x \geqslant 2$
and this can be written as
$\Rightarrow 2 \leqslant 2x \leqslant 4$
Now we divide the above inequality by 2 and we have
$\Rightarrow 1 \leqslant x \leqslant 2$
Therefore the domain of the function $y = {\sin ^{ - 1}}(3 - 2x)$ is $[1,2]$ and the range of the function $y = {\sin ^{ - 1}}(3 - 2x)$ is $[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} ]$.

Note:
The domain where the x values ranges and the range where the y values ranges. Here the function is a form of trigonometry function. We must know about the trigonometry and inverse trigonometry concept to solve this question. We should also know about the table for trigonometry ratios for the standard angles.