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Question
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 − x
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Solution
f : R → R, defined by f(x) = x3 − x
Injection test :
Let x and y be any two elements in the domain (R), such that f(x) = f(y).
f(x) = f(y)
x3−x=y3−y
Here, we cannot say x=y.
For example, x=1 and y=-1
x3−x =1−1= 0
y3−y=(−1)3−(−1)−1+1=0
So, 1 and -1 have the same image 0 .
So, f is not an injection.
Surjection test :
Let y be any element in the co-domain (R), such that f(x) = y for some element x in R (domain).
f(x) = y
x3 − x = y
By observation we can say that there exist some x in R, such that x3 - x = y.
So, f is a surjection and f is not a bijection.
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