Question

Which among the following has the largest domain.

• A. $y=cos\left(co{s}^{-1}\right(x\left)\right)$
• B. $y=cot\left(co{t}^{-1}\right(x\left)\right)$
• C. $y=sec\left(se{c}^{-1}\right(x\left)\right)$
• D. $y=cosec\left(cose{c}^{-1}\right(x\left)\right)$

Answer 1

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

Let’s take each function and see their domain and range. $y=cos\left(co{s}^{-1}\right(x\left)\right)$
We can see that $co{s}^{-1}\left(x\right)$ is defined only in the interval [-1, 1]. Same is true for $y=cos\left(co{s}^{-1}\right(x\left)\right)$ since cos x can take any real number as its argument.
Domain of $y=cos\left(co{s}^{-1}\right(x\left)\right):[-1,1]$
Range of $y=cos\left(co{s}^{-1}\right(x\left)\right):[-1,1]$
Similarly the domain and range of other functions can be found. Domain of $y=cot\left(co{t}^{-1}\right(x\left)\right):R$
Domain of $y=sec\left(se{c}^{-1}\right(x\left)\right):R-(-1,1)$
Domain of $y=cosec\left(cose{c}^{-1}\right(x\left)\right):R-(-1,1)$
From this we can see that the domain is largest for the function $f\left(x\right)=cot\left(co{t}^{-1}\right(x\left)\right)$.