If A = R - \2\ and B = R - \1\ and function f: A arrow B defined by f(x) = (x-1)/(x-2), then f is Option 1: one-one but not onto Option 2: onto but not one-one Option 3: one-one and onto Option 4: Neither one-one nor onto

Answer: one-one and onto. Solution: If f(x_1)=f(x_2), cross-multiplying gives x_1=x_2, so f is one-one. Solving y=(x-1)/(x-2) gives x=(2y-1)/(y-1), which is defined for every y≠1, so every element of B has a pre-image in A. Hence f is one-one and onto.

Algebra

Relations & Functions

Medium

core

If $A = R - \{2\}$ and $B = R - \{1\}$ and function $f: A \rightarrow B$ defined by $f(x) = \frac{x-1}{x-2}$ , then $f$ is

one-one but not onto

onto but not one-one

one-one and onto

Neither one-one nor onto

✅ Correct Option: 3

If $f(x_1)=f(x_2)$ , cross-multiplying gives $x_1=x_2$ , so $f$ is one-one. Solving $y=\frac{x-1}{x-2}$ gives $x=\frac{2y-1}{y-1}$ , which is defined for every $y\neq1$ , so every element of $B$ has a pre-image in $A$ . Hence $f$ is one-one and onto.

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Let $R : \mathbb{R} \to \mathbb{R}$ , where $\mathbb{R}$ is the set of real numbers and R be a relation. An element $(x, y) \in R$ if $x + y - \sqrt{2}$ belongs to the set of irrational numbers. Then the relation R is :

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Let $f : [2, \infty) \to \mathbf{R}$ be a function defined by $f(x) = x^2 - 4x + 5$ . The range of $f$ is :

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Let A = {1,2,3}. Consider the relation R = {(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}. Then R is

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If $A = \{1, 2, 3, 4, ..., n\}$ and $B = \{x, y\}$ , then the number of surjections from A to B is

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Which of the following relations on the set $A = \{1, 2, 3\}$ are equivalence ?

(A) $R = \{(1,1), (2,2), (1,2), (2,1)\}$

(B) $R = \{(1,1), (2,2), (3,3)\}$

(D) $R = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)\}$

(E) $R = \{(1,1), (2,2), (3,3), (1,2)\}$

Choose the correct answer from the options given below :

If A=R−{2}A = R - \{2\}A=R−{2} and B=R−{1}B = R - \{1\}B=R−{1} and function f:A→Bf: A \rightarrow Bf:A→B defined by f(x)=x−1x−2f(x) = \frac{x-1}{x-2}f(x)=x−2x−1​, then fff is

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CUET Mathematics Past Year Questions (Topic-Wise)

If $A = R - \{2\}$ and $B = R - \{1\}$ and function $f: A \rightarrow B$ defined by $f(x) = \frac{x-1}{x-2}$ , then $f$ is