Classify the following functions as injection, surjection or bije… - Vidyadip

Question

Classify the following functions as injection, surjection or bijection:$f : R \rightarrow R,$ defined by $f(x) = 1 + x^2$

✓

Answer

$f : R \rightarrow R,$ defined by $f(x) = 1 + x^2$
Injection test: Let $\text{x, y}\in\text{R,}$ such that,
$f(x) = f(y)$
$\Rightarrow 1 + x^2 = 1 + y^2$
$\Rightarrow x^2 - y^2 = 0$
$\Rightarrow (x - y)(x + y) = 0$
either $x = y$ or $x = -y$ or $\text{x}\neq\text{y}$
Therefore, $f$ is not one$-$one.
Surjection: Let $\text{y}\in\text{R}$ be arbitrary, then
$f(x) = y$
$\Rightarrow 1 + x^2 = y$
$\Rightarrow x^2 + 1 - y = 0$
$\therefore\ \text{x}\pm\sqrt{\text{y}-1}\notin\text{R}$ or $y < 1$
$\therefore f$ is not onto.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from RELATIONS AND FUNCTIONS→RELATIONS AND FUNCTIONS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\left( {2\vec a + \vec b} \right)$and$\left( {\vec a - 3\vec b} \right)$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.

A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

On Q, the set of all rational numbers a binary operation * is defined by $\text{a}\ ^*\ \text{b}=\frac{\text{a}+\text{b}}{2}.$ Show that * is not associative on Q.

A man wins a rupee for head and loses a rupee for tail when a coin is tossed. Suppose that he tosses once and quits if he wins but tries once more if he loses on the first toss. Find the probability distribution of the number of rupees the man wins.

Solve the following differential equation
$\text{x}\frac{\text{dy}}{\text{dx}}+\cot\text{y}=0$

Evaluate the following integrals:
$\int\tan^\frac{3}{2}\text{x}\sec^2\text{x dx}$

Solve the following $\ce{LPP}$ by graphical method:
Minimize $Z = 20x + 10y$
Subject to
$x+2 y \leq 40$
$3 x+y \geq 30$
$4 x+3 y \geq 60$
and $x, y \geq 0$

Represent the following families of curves by forming the corresponding differential equation:
$\text{y}=\text{ax}^3$

Write the position vector of the point which divdes the join of the points with position vectors $3\vec{\text{a}} - 2\vec{\text{b}} $ and $2\vec{\text{a}} + 3\vec{\text{b}} $ in the ratio 2 : 1.

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:

An injective map from A to B.
A mapping from A to B which is not injective.
A mapping from A to B.