Question

The function $f:R\to R$ is defined by $f\left(x\right)=3^{-x}$. Observe the following statements
I. $f$ is one-one.
II. $f$ is onto.
III. $f$ is a decreasing function.
Out of these, true statements are

• A. I, III
• B. II, III
• C. I, II
• D. All of these

Answer 1

The correct option is A I, III

Given, $f:R\to R$ such that $f\left(x\right)=3^{-x}$
Let $y_1$ and $y_2$ be two elements of $f\left(x\right)$ such that $y_1=y_2$

$\Rightarrow f\left(x_1\right)=f\left(x_2\right)$
$\Rightarrow 3^{-x_1}=3^{-x_2}$

$\Rightarrow 3^{x_1}=3^{x_2}$

$\Rightarrow x_1=x_2$

So, $f\left(x\right)$ is one-one.
Since, $f\left(x\right)$ is positive for every value of $x$, therefore $f\left(x\right)$ is into i.e. not onto.
Now, $f\left(x\right)=3^{-x}$

$\Rightarrow f^{\prime }\left(x\right)=-3^{-x}\log 3<0\forall x\in R$
$\therefore$ It is a decreasing function.
Hence, statements I and III are true.