Question

The function $f: R \rightarrow R$ is defined as $f(x)=3^{-x}$. From the following statements,
I. $f$ is one-one
II. $f$ is onto
III. $f$ is a decreasing function
the true statements are

• A. only I, II
• B. only II, III
• C. only I, III
• D. I, II, III

Answer 1

Answer: (C)

Since, $f: R \rightarrow R$ such that $f(x)=3^{-x}$
Let $y_{1}$ and $y_{2}$ be two elements of $f(x)$ such that $y_{1}=y_{2}$
$\Rightarrow 3^{-x_{1}}=3^{-x_{2}}$
$ \Rightarrow x_{1}=x_{2}$
Since, if two images are equal, then their elements are equal,
therefore it is one-one function.
Since, $f(x)$ is positive for every value of $x$,
therefore $f(x)$ is into.
On differentiating w.r.t. $x$,
we get $\frac{d y}{d x}=-3^{-x} \log 3< 0$ for every value of $x$.
$\therefore$ It is decreasing function.
$\therefore$ Statement I and III are true.