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Classify the Following Functions as Injection, Surjection Or Bijection : F : Z → Z, Defined By F(X) = X2 + X - CBSE (Arts) Class 12 - Mathematics

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Question

Classify the following function as injection, surjection or bijection :

f : Z → Z, defined by f(x) = x2 + x

Solution

  f : Z → Z, defined by f(x) = x2 + x

Injection test:

Let x and y be any two elements in the domain (Z), such that f(x) = f(y).

f(x= f(y)

x2y+ y

Here, we cannot say that x = y.

For example, x = 2 and y = - 3

 Then,

x2+x=22+2= 6

y2+y=(3)23= 6

So, we have two numbers 2 and -3 in the domain Z whose image is same as 6.

So, f is not an injection .

Surjection test:

Let y be any element in the co-domain (Z), such that f(x) = y for some element x in Z (domain).

f(x) = y

x2 y

Here, we cannot say ∈ Z.

For example, y = - 4.

x2 − 4

x20

=` (-1 ±sqrt-5)/2 = (-1 ±isqrt5)/2`  which is not in Z.

So, f is not a surjection and  f is not a bijection.

  Is there an error in this question or solution?
Solution Classify the Following Functions as Injection, Surjection Or Bijection : F : Z → Z, Defined By F(X) = X2 + X Concept: Types of Functions.
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