Let R denote the set of all real numbers and | vedclass

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

Question

(D) The function is defined as $f(x) = x^2$.$A$. $f$ is one-one and onto, if $A = B = R^{+}$.For $A = R^{+}$, $f(x) = x^2$ is strictly increasing, so

Vedclass · Accepted Answer

Vedclass · Answer

(D) The function is defined as $f(x) = x^2$.$A$. $f$ is one-one and onto, if $A = B = R^{+}$.For $A = R^{+}$, $f(x) = x^2$ is strictly increasing, so it is one-one. Since $B = R^{+}$, for every $y \in R^{+}$, there exists $x = \sqrt{y} \in R^{+}$ such that $f(x) = y$, so it is onto. Thus, $A \rightarrow 4$.$B$. $f$ is one-one but not onto, if $A = R^{+}, B = R$.For $A = R^{+}$, $f(x) = x^2$ is one-one. Since $B = R$, the range is $R^{+}$, which is a proper subset of $B$, so it is not onto. Thus, $B \rightarrow 1$.$C$. $f$ is onto but not one-one, if $A = R, B = R^{+}$.For $A = R$, $f(1) = 1$ and $f(-1) = 1$, so it is not one-one. Since $B = R^{+}$, for every $y \in R^{+}$, there exists $x = \pm \sqrt{y} \in R$ such that $f(x) = y$, so it is onto. Thus, $C \rightarrow 3$.$D$. $f$ is neither one-one nor onto, if $A = B = R$.For $A = R$, $f(1) = f(-1) = 1$, so it is not one-one. Since $B = R$, the range is $R^{+}$, which is a proper subset of $B$, so it is not onto. Thus, $D \rightarrow 2$.Therefore, the correct matching is $A \rightarrow 4, B \rightarrow 1, C \rightarrow 3, D \rightarrow 2$.

Similar Questions

Let $f(x) = x^{12} - x^9 + x^4 - x + 1$. Which of the following is true?

The number of roots of the equation $\sqrt{2}+e^{\cosh^{-1} x}-e^{\sinh^{-1} x}=0$ is

Function $f(x) = \sin x + \tan x + \operatorname{sgn}(x^2 - 6x + 10)$ is (where $\operatorname{sgn}$ is the signum function):

Let $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements in the relation $R = \{((a_1, b_1), (a_2, b_2)) \in ((A \times B) \times (A \times B)) : a_1 + a_2 \text{ divides } b_2 + b_1\}$ is . . . . . . .

Let $S = \{1, 2, 3, 4, 5, 6, 7\}$. Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in S$ and $m \cdot n \in S$ is equal to $......$

Vedclass Test Series

Exam Paper Generator

Online Exam Module