Question

Let $f$ be an injective map with domain {x, y, z} and range {1, 2, 3} such that exactly one of the following statements is correct and the remaining are false :
$f\left(x\right)=1,f\left(y\right)\sqrt{1},f\left(z\right)\sqrt{2}$. The value of $f^{-1}\left(1\right)$ is

• A. x
• B. y
• C. z
• D. none of these

Answer 1

Answer: (B)

y

$f\left(x\right)=1,f\left(y\right)\ne 1,f\left(z\right)\ne 1$

Case 1:

$f\left(x\right)=1f\left(z\right)=2f\left(y\right)=1$

$\therefore f$ is not injective

Case 2: $f\left(y\right)\ne 1,f\left(z\right)=2,f\left(x\right)=1$

Case 3:

$f\left(z\right)\ne 2f\left(z\right)=3f\left(x\right)\ne 1f\left(x\right)=2f\left(y\right)=1f\left(y\right)=1f\left(x\right)=2,f\left(y\right)=1,f\left(z\right)=3f^{-1}\left(2\right)=x,f^{-1}\left(1\right)=y,f^{-1}\left(3\right)=z$