Let f: R R be defined as f(x) = 3x. Choose the correct answer. - Vidyadip

MCQ

Let $f: R \rightarrow R$ be defined as $f(x) = 3x.$ Choose the correct answer.

✓
$f$ is one$-$one onto
B
$f$ is many$-$one onto
C
$f$ is one$-$one but not onto
D
$f$ is neither one$-$one nor onto.

✓

Answer

Correct option: A.

$f$ is one$-$one onto

$f: R \rightarrow R$ is defined as $f(x) = 3x.$
Let $\text{x},\text{y}\in\text{R}$ such that $f(x) = f(y)$.
$\Rightarrow 3x = 3y$
$ \Rightarrow x = y$
$\therefore f$ is one$-$one.
Also, for any real number $(y)$ in $co-$domain $R,$ there exists $\frac{\text{y}}{3}$ in $R$
such that $f\Big(\frac{\text{y}}{3}\Big)=3\Big(\frac{\text{y}}{3}\Big)=\text{y}$
$\therefore f$ is onto.
Hence, function $f$ is one$-$one and onto.
The correct answer is $A.$

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