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

Question

Classify the following functions as injection, surjection or bijection:$f : R \rightarrow R$, defined by$ f(x) = 5x^3 + 4$

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Answer

f : R → R, defined by $f(x) = 5x^3 + 4$
Injection test: Let x and y be any two elements in the domain (R), such that f(x) = f(y).
$f(x) = f(y)$
$5x^3 + 4 = 5y^3 + 4$
$5x^3 = 5y^3$
$x^3 = y^3$
$x = y$
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).
$f(x) = y$
$5x^3 + 4 = y$
$5x^3 = y - 4$
$\text{x}^3=\frac{\text{y}-4}{5}$
$\text{x}=\sqrt[3]{\frac{\text{y}-4}{5}}\in\text{R}$
So, f is a surjection and f is a bijection.

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