Classify the following functions as injection, surjection or bije… - Vidyadip

Question

Classify the following functions as injection, surjection or bijection:
f : R → R, defined by f(x) = x³ + 1

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Answer

f : R → R, defined by f(x) = x³ + 1
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 + 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
$\text{x}=\sqrt[3]{\text{y}-1}\in\text{R}$
So, f is a surjection.
So, f is a bijection.

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