Write the domain for the function $f\left( x \right)={{\cos }^{-1}}\left( x \right)$?

Hint: We start solving the problem by assuming that the given function is equal to y, we then make the necessary arrangements to find the function for ‘x’. We then recall the property of the cosine function that the range of cosine functions is $\left[ -1,1 \right]$ which will be values of x. We then recall that these are values that the given function will be defined which in turn are the elements present in the domain of the function $f\left( x \right)={{\cos }^{-1}}\left( x \right)$.

Complete step by step answer:
According to the problem, we need to find the domain of the function $f\left( x \right)={{\cos }^{-1}}\left( x \right)$.
Let us assume $f\left( x \right)=y$.
So, we get $y={{\cos }^{-1}}\left( x \right)$.
$\Rightarrow \cos y=x$.
We know that the range of any cosine function is $\left[ -1,1 \right]$.
So, we have $\left[ -1,1 \right]=x$.
$\Rightarrow x\in \left[ -1,1 \right]$.
 So, the given function $f\left( x \right)={{\cos }^{-1}}\left( x \right)$ is defined for $x\in \left[ -1,1 \right]$ which will be the domain.

∴ The domain of the function $f\left( x \right)={{\cos }^{-1}}\left( x \right)$ is $\left[ -1,1 \right]$.

Note: We can solve this problem by drawing the plot of the cosine function $\cos y$ and checking the values of x. We should not confuse the domain of the function with the range of the range of the function. Whenever we get problems involving the domain of the inverse trigonometric functions, we try to equate it to a variable and may use the range of the trigonometric function related to that inverse function. Similarly, we can expect problems to find the range of the function $f\left( x \right)={{\cos }^{-1}}\left( x \right)$.