Question

If $g$ is the inverse function of $f$ and $f^1\left(x\right)=\sin x$ then $g^1\left(x\right)$ is

• A. $\csc \left(g\right(x\left)\right)$
• B. $\sin \left(g\right(x\left)\right)$
• C. $-\frac{1}{\sin \left(g\right(x\left)\right)}$
• D. $\cos \left(g\right(x\left)\right)$

Answer 1

The correct option is A $\csc \left(g\right(x\left)\right)$

$g\left(x\right)=f^{-1}\left(x\right)$

$f^{\prime }\left(x\right)=sinx,g^{\prime }\left(x\right)=?$

$f\left(g\right(x\left)\right)=x.$

$f^{\prime }\left(g^{\prime }\right(x\left)\right).g^{\prime }\left(x\right)=1$

$sin\left(g\right(x\left)\right)=1$

$g^{\prime }\left(x\right)sin\left(g\right(x\left)\right)=1$

$g^{\prime }\left(x\right)=1/sin\left(g\right(x\left)\right)$

$g^{\prime }\left(x\right)=cosec\left(g\right(x\left)\right)$

option (a) is correct.