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The function $f: R \rightarrow R$ is defined by $f(x)=3^{-x}$. Observe the following statements of it
I. $f$ is one-one
II. $f$ is onto
III. $f$ is a decreasing function
Out of these, true statement are
Options:
I. $f$ is one-one
II. $f$ is onto
III. $f$ is a decreasing function
Out of these, true statement are
Solution:
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Verified Answer
The correct answer is:
Only I, II
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
$$
\begin{aligned}
& y_1=y_2 \\
& \Rightarrow \quad 3^{-x_1}=3^{-x_2} \Rightarrow x_1=x_2 \\
&
\end{aligned}
$$
Since, if two images are equal, then their elements are equal, therefore it is one-one function
Let
$$
y=3^{-x}
$$
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 II are true.
Let $y_1$ and $y_2$ be two elements of $f(x)$ such that
$$
\begin{aligned}
& y_1=y_2 \\
& \Rightarrow \quad 3^{-x_1}=3^{-x_2} \Rightarrow x_1=x_2 \\
&
\end{aligned}
$$
Since, if two images are equal, then their elements are equal, therefore it is one-one function
Let
$$
y=3^{-x}
$$
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 II are true.
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