Question

Let $f$ be a function on $R$ given by $f\left(x\right)=x^2$ and let $E=\left\{x\in R:-1\le x\le 0\right\}$ and $F=\left\{x\in R:0\le x\le 1\right\}$ then which of the following is false?

• A. $f\left(E\right)=f\left(F\right)$
• B. $E\cap F\subset f\left(E\right)\cap f\left(F\right)$
• C. $E\cup F\subset f\left(E\right)\cup f\left(F\right)$
• D. $f\left(E\cap F\right)=0$

Answer 1

Answer: (B), (C)

$E\cup F\subset f\left(E\right)\cup f\left(F\right)$

Explanation for correct option:

Given, the functions are:

$f\left(x\right)=x^2$

$E=\left\{x\in R:-1\le x\le 0\right\}$

$F=\left\{x\in R:0\le x\le 1\right\}$

Consider the given equation as,

$$\begin{gathered} f\left(E\right)=\left\{x\in R:0\le x^2\le 1\right\}.....\left(1\right) \\ =\left[0,1\right] \end{gathered}$$

$$\begin{gathered} f\left(F\right)=\left\{x\in R:0\le x^2\le 1\right\}....\left(2\right) \\ =\left[0,1\right] \end{gathered}$$

Checking for option (B).

$\mathrm{E}\cap \mathrm{F}$ is the intersection of $\mathrm{E}$and $\mathrm{F}$.

whereas $f\left(E\right)\cap f\left(F\right)$ is the intersection of $f\left(E\right)andf\left(F\right)$.

Now,

$\mathrm{E}\cap \mathrm{F}=\left\{0\right\}$

$f\left(E\right)\cap f\left(F\right)=\left[0,1\right]$

Therefore, $E\cap F\subset f\left(E\right)\cap f\left(F\right)$ is false.

Checking for option (C).

From the Equation (A) and (B).

$$\begin{gathered} \mathrm{E}\cup \mathrm{F}=\left[-1,1\right] \\ \mathrm{f}\left(\mathrm{E}\right)\cup \mathrm{f}\left(\mathrm{F}\right)=\left[0,1\right] \end{gathered}$$

Therefore, $E\cup F\subset f\left(E\right)\cup f\left(F\right)$ is false.

Hence, option (C) is incorrect.

Hence, the correct answer is option (B)

Explanation for incorrect option

Checking for option (A).

Now, it is evident that $f\left(E\right)=f\left(F\right)=\left[0,1\right]$.

So, Option (A) is true.

Checking for option (D).

From the equation $\left(1\right)\&\left(2\right)$:

$f\left(E\cap F\right)=f\left(0\right)=0^2=0$

Hence, $f\left(E\cap F\right)=0$ holds true.

Hence, the correct answer is Option (B) and (C)