# If f :

Question:

If $t: \mathrm{Q} \rightarrow \mathrm{Q}$ is defined as $f(x)=x^{2}$, then $f^{-1}(9)$ is equal to

(a) 3

(b) −3

(c) {−3, 3}

(d) ϕ

Solution:

(c) {−3, 3}

If $f: A \rightarrow B$, such that $y \in B$, then $f^{-1}\{y\}=\{x \in A: f(x)=y\}$.

In other words, $f^{-1}\{y\}$ is the set of pre-images of $y$.

Let $f^{-1}\{9\}=x$

Then, $f(x)=9$

$\Rightarrow x^{2}=9$

$\Rightarrow x=\pm 3$

$\therefore f^{-1}\{9\}=\{-3,3\}$