7. Determine whether each of these functions is a bijection from R to R.a) f (x) = −3x + 4 b) f (x) = −3x2 + 7 c) f (x) = (x + 1)/(x + 2) d) f (x) = x5 + 1
Question
7. Determine whether each of these functions is a bijection from R to R.
a) f (x) = −3x + 4
b) f (x) = −3x^2 + 7
c) f (x) = (x + 1)/(x + 2)
d) f (x) = x^5 + 1
Solution
a) f(x) = -3x + 4 is a bijection. It is both injective and surjective. It is injective because for every x1 ≠ x2, f(x1) ≠ f(x2). It is surjective because for every y in R, there exists an x in R such that f(x) = y.
b) f(x) = -3x^2 + 7 is not a bijection. It is not injective because for x1 ≠ x2, f(x1) can be equal to f(x2). For example, f(1) = f(-1). It is not surjective because for y > 7, there does not exist an x in R such that f(x) = y.
c) f(x) = (x + 1)/(x + 2) is not a bijection. It is injective because for every x1 ≠ x2, f(x1) ≠ f(x2). However, it is not surjective because
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