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Question

A function f : R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).

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Solution

Injectivity of f:
Let x and y be two elements of domain (R), such that
fx=fyx3+4=y3+4x3=y3x=y
So, f is one-one.

Surjectivity of f:
Let y be in the co-domain (R), such that f(x) = y.

x3+4=yx3=y-4x=y-43R domain

f is onto.
So, f is a bijection and, hence, is invertible.

Finding f -1:
Let f-1x=y ...1x=fyx=y3+4x-4=y3y=x-43So, f-1x=x-43 [from 1]f-13=3-43 =-13=-1

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