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Question
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4
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Solution
f : R → R, defined by f(x) = 5x3 + 4
Injection test :
Let x and y be any two elements in the domain (R), such that f(x) = f(y).
f(x) = f(y)
5x3+4 = 5y3+4
5x3= 5y3
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
5x3+4 = y
5x3+ 4 = y
`x^3 = (y - 4)/5 ∈ R `
So, f is a surjection and f is a bijection.
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