Find Pre-Images of a Function
Question:
If
$$
f:\mathbb{R}\to\mathbb{R}
$$
is defined by
$$
f(x)=x^2+1
$$
then find
$$
f^{-1}\{17\}
$$
and
$$
f^{-1}\{-3\}
$$
Solution
Given: $$ f(x)=x^2+1 $$
Find \(f^{-1}\{17\}\)
$$ x^2+1=17 $$ $$ x^2=16 $$ $$ x=\pm4 $$
Therefore, $$ f^{-1}\{17\}=\{-4,4\} $$
Find \(f^{-1}\{-3\}\)
$$ x^2+1=-3 $$ $$ x^2=-4 $$
No real number satisfies this equation.
Therefore, $$ f^{-1}\{-3\}=\phi $$