In which of the function is one-one defined in R R . - Vidyadip

MCQ

In which of the function is one-one defined in $R \rightarrow$ R .

A
$f(x)=|x|$
B
$f(x)=\cos x$
✓
$f(x)=e^x$
D
$f(x)=x^2$.

✓

Answer

Correct option: C.

$f(x)=e^x$

(C)$f(x)=e^x$
$x_1, x_2 \in R$ such that
$f\left(x_1\right)=f\left(x_2\right)$
$\Rightarrow \quad e^{x_1}=e^{x_2}$
$\Rightarrow \quad \log _e e^{x_1}=\log _e e^{x_2}$
$\Rightarrow \quad x_1 \log _e e=x_2 \log _e e$
$\Rightarrow \quad x_1=x_2 \quad\left[\because \log _e e=1\right]$
so, $\quad f\left(x_1\right)=f\left(x_2\right)$
$\Rightarrow \quad x_1=x_2 \quad \forall x_1, x_2 \quad R$
$\therefore f$ is one-one function

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