Question

The function $f:R\to R$ 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, $f:R\to R$ 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 $\frac{dy}{dx}=-3^{-x}\log 3<0$ for every value of $x$.
$\therefore$ It is decreasing function.
$\therefore$ Statement I and III are true.