Let f(x) = x2 , x [ ] R.

Question

Let f(x) = x2
, x $$ \in $$ R. For any A $$ \subseteq $$ R, define g (A) = { x $$ \in $$ R : f(x) $$ \in $$ A}. If S = [0,4], then which one of the
following statements is not true ?

Accepted Answer

f(x) = x2    x $$ \in $$  R

g(A) = {x $$ \in $$ R : f(x) $$ \in $$ A} S $$ \equiv $$ [0, 4]

g(S) = {x  $$ \in $$ R : f(x)  $$ \in $$ S}

= {x $$ \in $$ R : 0 $$ \le $$ x2 $$ \le $$ 4}

= {x $$ \in $$ R : --2 $$ \le $$ x $$ \le $$ 2}

$$ \therefore $$ g(S) $$ \ne $$ S

$$ \therefore $$ f(g(S)) $$ \ne $$ f(S)

g(f(S)) = {x $$ \in $$ R : f(x) $$ \in $$ f(S)}

= {x $$ \in $$ R : x2 $$ \in $$ S2}

= {x $$ \in $$ R : 0 $$ \le $$ x2 $$ \le $$ 16}

= {x $$ \in $$ R : --4  $$ \le $$ x  $$ \le $$ 4}

$$ \therefore $$ g(f(S)) $$ \ne $$ g(S)

$$ \therefore $$ g(f(S)) = g(S) is incorrect

Solutions · 1