Question

Consider the following functions:
$1.f\left(x\right)=x^3,xϵR$
$2.f\left(x\right)=\sin x,-\pi /2<x<\pi /2$
$3.f\left(x\right)=e^x,xϵR$
Which of the above functions have inverse defined on their ranges?

• A. $1$ and $2$ only
• B. $2$ and $3$ only
• C. $1$ and $3$ only
• D. $1,2$ and $3$

Answer 1

Answer: (A)

$1$ and $2$ only

For case (1), Let $y=x^3⟹x=y^{\frac{1}{3}}$ which is defined on the $R$ range.

For case (2), Let $y=\sin x⟹x={\sin }^{-1}y$ which is defined on the $(-\frac{\pi }{2},\frac{\pi }{2})$ range.

For case (3), Let $y=e^x⟹x=\ln y$ which cannot be defined on $R$ as the only inputs it can take are positive.