Let f: X rightarrow X be such that f(f(x)) = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given that $f(f(x)) = x$ for all $x \in X$. To check for one-to-one (injective): Let $f(x_1) = f(x_2)$. Applying $f$ on both sides, we get $f(f(x_

Vedclass · Accepted Answer

$f$ is both one-to-one and onto

Vedclass · Answer

(D) Given that $f(f(x)) = x$ for all $x \in X$. To check for one-to-one (injective): Let $f(x_1) = f(x_2)$. Applying $f$ on both sides, we get $f(f(x_1)) = f(f(x_2))$. Since $f(f(x)) = x$, this implies $x_1 = x_2$. Thus, $f$ is one-to-one. To check for onto (surjective): For any $y \in X$, let $x = f(y)$. Then $f(x) = f(f(y)) = y$. Since for every $y \in X$, there exists an $x \in X$ such that $f(x) = y$, $f$ is onto. Therefore, $f$ is both one-to-one and onto (bijective).

Let $f: X \rightarrow X$ be such that $f(f(x)) = x$ for all $x \in X$ and $X \subseteq \mathbb{R}$. Then:

Similar Questions

$f: N \rightarrow N$,is defined by $f(x)=x^6$ then, . . . . . . .

The function $f:[0, \infty) \rightarrow [0, \infty)$ defined by $f(x) = \frac{x}{1+x}$ is

If a real-valued function $f:[a, \infty) \rightarrow [b, \infty)$ defined by $f(x) = 2x^2 - 3x + 5$ is a bijection,then $3a + 2b =$

Define $f: R \rightarrow R$ by $f(x) = \max \{x+1, 1-x, 2\}$. Then,$f$ is

If $f: N \rightarrow R$ is defined by $f(1)=-1$ and $f(n+1)=3f(n)+2$ for $n \geq 1$, then $f$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module