Functions 2 Question 1
2. Let $f(x)=x^{2}, x \in R$. 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?
(2019 Main, 10 April I)
(a) $f(g(S))=S$
(b) $g(f(S)) \neq S$
(c) $g(f(S))=g(S)$
(d) $f(g(S)) \neq f(S)$
Show Answer
Answer:
Correct Answer: 2. (c)
Solution:
- Given, functions $f(x)=x^{2}, x \in R$ and $g(A)={x \in R: f(x) \in A} ; A \subseteq R$
Now, for $S=[0,4]$
$$ \begin{aligned} & g(S)={x \in R: f(x) \in S=[0,4]} \\ & ={x \in R: x^{2} \in[0,4] } \\ & ={x \in R: x \in[-2,2]} \\ & \Rightarrow \quad g(S)=[-2,2] \\ & \text { So, } \quad f(g(S))=[0,4]=S \\ & \text { Now, } f(S)={x^{2}: x \in S=[0,4] }=[0,16] \\ & \text { and } g(f(S))={x \in R: f(x) \in f(S)=[0,16]} \\ & ={x \in R: f(x) \in[0,16]} \\ & ={x \in R: x^{2} \in[0,16] } \\ & ={x \in R: x \in[-4,4]}=[-4,4] \end{aligned} $$
From above, it is clear that $g(f(S))=g(S)$.
