Question

Which of the following functions satisfy the condition $f\left(x\right)=f-1\left(x\right)$?
I) $f\left(x\right)=-x$
II) $f\left(x\right)=\sqrt{x}$
III) $f\left(x\right)=-1/x$

• A. III and II only
• B. III and I only
• C. III only
• D. II only
• E. I only

Answer 1

The correct option is B III and I only

For first part

$f\left(x\right)=y=-x$

when relacing $y$ by $x$, we get

$x=-y$

thus $y=-x=f^{-1}$

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

For second part

$y=\sqrt{x}$

when relacing $y$ by $x$, we get

$x=\sqrt{y}$

thus $y=x^2=f^{-1}$

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

For third part

$f\left(x\right)=y=-\frac{1}{x}$

when relacing $y$ by $x$, we get

$x=-\frac{1}{y}$

thus $y=-\frac{1}{x}=f^{-1}$

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

so option B is correct