Question

$y=$ has the largest domain.

• A. $\cos \left({\cos }^{-1}\right(x\left)\right)$
• B. $\cot \left({\cot }^{-1}\right(x\left)\right)$
• C. $\sec \left({\sec }^{-1}\right(x\left)\right)$
• D. $\sin \left({\sin }^{-1}\right(x\left)\right)$

Answer 1

The correct option is B $\cot \left({\cot }^{-1}\right(x\left)\right)$

Let’s take each function and see their domain and range.
$y=\cos \left({\cos }^{-1}\right(x\left)\right)$
We can see that ${\cos }^{-1}\left(x\right)$ is defined only in the interval [-1, 1]. Same is true for $y=\cos \left({\cos }^{-1}\right(x\left)\right)$ since $\cos x$ can take any real number as its argument.
Domain of $y=\cos \left({\cos }^{-1}\right(x\left)\right):[-1,1]$
Range of $y=\cos \left({\cos }^{-1}\right(x\left)\right):[-1,1]$
Similarly, the domain and range of other functions can be found.
Domain of $y=\cot \left({\cot }^{-1}\right(x\left)\right):R$
Domain of $y=\sec \left({\sec }^{-1}\right(x\left)\right):R-(-1,1)$
Domain of $y=\sin \left({\sin }^{-1}\right(x\left)\right):[-1,1]$
From this we can see that the domain is largest for the function $f\left(x\right)=\cot \left({\cot }^{-1}\right(x\left)\right)$.