The function f: R R defined as f(x)=x^3 is - Vidyadip

MCQ

The function $f: R \rightarrow R$ defined as $f(x)=x^3$ is

A
One-one but not onto
B
Not one-one but onto
C
Neither one-one nor onto
D
One-one and onto

✓

Answer

Let $x_1, x_2 \in R$ be such that $f\left(x_1\right)=f\left(x_2\right)$
$\Rightarrow x_1^3=x_2^3 \Rightarrow x_1=x_2 \Rightarrow f$ is one-one.
Let $f(x)=x^3=y$ for some arbitrary element $y \in R \Rightarrow x=y^{1 / 3}$
$\Rightarrow f\left(y^{1 / 3}\right)=y$
Every image $y \in R$ has a unique pre-image in $R$.
$\Rightarrow f$ is onto
$\therefore \quad f$ is one-one and onto.

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