Algebra

Relations & Functions

Easy

core

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 100x + 1$ , where $\mathbb{R}$ is a set of real numbers, then

$f$ is one-one but not onto

$f$ is onto but not one-one

$f$ is both one-one and onto

$f$ is neither one-one nor onto

✅ Correct Option: 3

To determine if $f(x) = 100x + 1$ is one-one (injective), assume $f(x_1) = f(x_2)$ :

$100x_1 + 1 = 100x_2 + 1$

$100x_1 = 100x_2$

$x_1 = x_2$

Since $f(x_1) = f(x_2)$ implies $x_1 = x_2$ , different inputs produce different outputs.

Therefore, $f$ is one-one.

To determine if $f(x) = 100x + 1$ is onto (surjective), let $y$ be any real number.

Find $x$ such that $f(x) = y$ :

$100x + 1 = y$

$100x = y - 1$

$x = \frac{y - 1}{100}$

For any real number $y$ , the value $x = \frac{y - 1}{100}$ is a real number and satisfies $f(x) = y$ .

Therefore, $f$ is onto.

Since $f$ is both one-one and onto, $f$ is a bijection.

The function $f(x) = 100x + 1$ is both one-one and onto.

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For the relation $R = \{(a, b): a \leq b\}$ in $\mathbb{R}$ , which of the following is correct?

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Match the LIST-I with LIST-II

[Where R and Z denotes set of real numbers and set of integers respectively]

	LIST-I Function		LIST-II Nature
A.	$f: R \to R, f(x) = \sin^2 x + \cos^2 x$	I.	Bijective
B.	$f: Z \to Z, f(x) = x^3$	II.	Surjective but not injective
C.	$f: R \to R, f(x) = x^3 - x$	III.	Injective but not surjective
D.	$f: R \to R, f(x) = 3 - 4x$	IV.	Neither injective nor surjective

Choose the correct answer from the options given below:

Let f:R→Rf: \mathbb{R} \to \mathbb{R}f:R→R be defined as f(x)=100x+1f(x) = 100x + 1f(x)=100x+1, where R\mathbb{R}R is a set of real numbers, then

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 100x + 1$ , where $\mathbb{R}$ is a set of real numbers, then