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Question
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 + 1
Solution
f : R → R, defined by f(x) = x3 + 1
Injection tes:
Let x and y be any two elements in the domain (R), such that f(x) = f(y).
f(x) = f(y)
x3+1 = y3+ 1
x3=y3
x = y
So, f is 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+1=y
` x = 3sqrt (y - 1)∈ R `
So, f is a surjection.
So, f is a bijection.
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