Algebra

Relations & Functions

Easy

core

Let $f: \mathbb{R}$ -> $\mathbb{R}$ be a function defined as $f(x) = x^4$ . Which one of the following is true?

$f$ is one-one and onto

$f$ is one-one but not onto.

$f$ is onto but not one-one.

$f$ is neither one-one nor onto.

✅ Correct Option: 4

The function $f(x) = x^4$ maps real numbers to real numbers. Two properties need to be checked:

One-one (Injective): Each output comes from only one input
Onto (Surjective): Every real number can be obtained as an output

A function is one-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$ .

Consider $f(2)$ and $f(-2)$ :

$f(2) = 2^4 = 16$

$f(-2) = (-2)^4 = 16$

Since $f(2) = f(-2)$ but $2 \neq -2$ , the function is not one-one.

A function $f: \mathbb{R} \to \mathbb{R}$ is onto if every real number can be obtained as an output.

For $x^4$ with any real number $x$ :

When $x = 0$ : $f(0) = 0^4 = 0$

When $x = 2$ : $f(2) = 2^4 = 16$

When $x = -2$ : $f(-2) = (-2)^4 = 16$

Since the power is even, $x^4 \geq 0$ for all real $x$ .

To obtain $y = -1$ as output, we would need $x^4 = -1$ , which has no real solution.

The range of $f$ is $[0, \infty)$ , not all of $\mathbb{R}$ .

Since negative numbers cannot be outputs, $f$ is not onto.

The function $f$ is neither one-one nor onto.

Not one-one because $f(2) = f(-2) = 16$
Not onto because negative numbers cannot be outputs

2025: 13 May Shift 2

Let L be the set of all lines in a plane and R be the relation on set L defined by $R = \{(L_1, L_2): L_1 \perp L_2\}$ Then R is

(A) an equivalence Relation

(B) a symmetric Relation

(D) a reflexive Relation

Choose the correct answer from the options given below:

2025: 13 May Shift 2

Match List-I with List-II

Let $f: A \rightarrow B$ be a function given by $f(x) = x^2$

List-I	List-II
Domain and Co-domain	Kind
(A) $A = \mathbb{R}$ and $B = \mathbb{R}$	(I) $f$ is both one-one and onto
(B) $A = \mathbb{R}$ and $B = [0, \infty]$	(II) $f$ is one-one but not onto
(C) $A = B = [0, \infty]$	(III) $f$ is not one-one but onto
(D) $A = [0, \infty]$ and $B = \mathbb{R}$	(IV) $f$ is neither one-one nor onto

Choose the correct answer from the options given below:

2025: 29 May Shift 2

Assume that R is a relation on the set Z of integers and it is given by $(x, y) \in R \Leftrightarrow |x - y| \leq 1$ . Then, R is

2025: 15 May Shift 1

Let A= {1, 2, 3}, then the possible equivalence relations on A are:

(A) {(1, 1), (2, 2) , (3, 3)}

(B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}

(D) {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)}

Choose the correct answer from the options given below:

2026: 26 May Shift 2

Let $A = \{ 1,2,3 \}$ , then Match List-I with List-II

LIST-I	LIST-II
A. $R_1 = \{(1,1), (2,2), (3,3), (1,2)\}$	I. Symmetric but neither reflexive nor transitive
B. $R_2 = \{(1,1), (3,3), (1,3), (3,1)\}$	II. Reflexive and transitive but not symmetric
C. $R_3 = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)\}$	III. Symmetric and transitive but not reflexive
D. $R_4 = \{(2,2), (3,3), (1,2), (2,1)\}$	IV. Reflexive and symmetric but not transitive

Choose the correct answer from the options given below:

2026: 30 May Shift 2

The greatest integer function $f: R\to R$ defined by $f(x) = [x]$ , then which of the following are true?

(A) $f(x)$ is one - one and onto.

(B) Domain of $f(x) = R$ (the set of real numbers).

(D) Range of $f(x) = Z$ (The set of integers).

Choose the correct answer from the options given below:

Let f:Rf: \mathbb{R}f:R -> R\mathbb{R}R be a function defined as f(x)=x4f(x) = x^4f(x)=x4. Which one of the following is true?

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

Let $f: \mathbb{R}$ -> $\mathbb{R}$ be a function defined as $f(x) = x^4$ . Which one of the following is true?