Question

How do you find the range and domain of $y={{\sin }^{-1}}\left( {{x}^{2}} \right)$?

Answer 1

From the question given, we have been asked to find the range and domain of $y={{\sin }^{-1}}\left( {{x}^{2}} \right)$. We can find the range and domain of the given function by knowing some conditions which satisfy the given function. We can clearly observe that the given function is an inverse sine function. We know that the inverse sine function is only defined for $-1\le x\le 1$ (Since sine is only taken for those values).

Complete step by step answer:
For answering this question we need to find the domain and range of the given function.
By observing we can surely say that it is the inverse sine function and we know that from the basic concept that the function is only defined for $-1\le x\le 1$ .
So, we know that our given function will only be defined when ${{x}^{2}}$ also obeys the same interval.
We figure out when this is true by solving the following inequality: $-1\le {{x}^{2}}\le 1$
The first one, $-1\le {{x}^{2}}$ is true for all real numbers, $x\in R$
And we can solve the second one by taking square root on both sides:
$\Rightarrow {{x}^{2}}\le 1$
$\Rightarrow \sqrt{{{x}^{2}}}\le \sqrt{1}$
$\Rightarrow x\le 1$
Now, we combine this with our original interval, which we do by taking the most restrictive of the bounds: \left\\{ \text{-1}\le \text{x}\le \text{1} \right\\}
Therefore, Domain for the given function is \left\\{ \text{-1}\le \text{x}\le \text{1} \right\\}
Range of the function: The inverse sine function usually ranges from $-\dfrac{\pi }{2}$to$\dfrac{\pi }{2}$, but since negative values are only produced by negative inputs to the function, the ${{x}^{2}}$ makes it so that we will only ever get positive or zero values.
This means our range will be: \left\\{ 0\le y\le \dfrac{\pi }{2} \right\\}
Therefore, we got the range and domain for the given inverse trigonometric function.

Note: We should be well aware of the inverse trigonometric functions. Also, we should be well known about the process of finding range and domain of an inverse trigonometric function. Also, we should be very careful while finding the domain and range because intervals are quite confusing. Similarly we can find the range and domain of the inverse cosine function $y={{\cos }^{-1}}\left( x \right)$ so its domain and range will be given as $\left[ -1,1 \right]$ and $\left[ \dfrac{-\pi }{2},\dfrac{\pi }{2} \right]$ respectively.