Question
Classify the following functions as injection, surjection or bijection:
f : Z → Z, defined by f(x) = x2 + x

Answer

f : Z → Z, given 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)
x2 + x = y2 + y
Here, we cannot say that x = y.
For example, x = 2 and y = -3
Then, x2 + x = 22 + 2 = 6
y2 + y = (-3)2 - 3 = 6
So, we have two numbers 2 and -3 in the domain Z whose image is same as 6.
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 + x = y
Here, we cannot say $\text{x}\in\text{Z}$
For example, y = -4
x2 + x = -4
x2 + x + 4 = 0
$\text{x}=\frac{-1\pm\sqrt{-15}}{2}=\frac{-1\pm\text{i}\sqrt{15}}{2}$ which is not in Z.
So, f is not a surjection and f is not a bijection.

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