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(C) Given the function $f: R \rightarrow R$ defined by $f(x)=3^{-x}$.$I$. For one-one check: Let $f(x_1) = f(x_2)$.$3^{-x_1} = 3^{-x_2} \Rightarrow -x

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Only $I, III$

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(C) Given the function $f: R ightarrow R$ defined by $f(x)=3^{-x}$.$I$. For one-one check: Let $f(x_1) = f(x_2)$.$3^{-x_1} = 3^{-x_2} \Rightarrow -x_1 = -x_2 \Rightarrow x_1 = x_2$.Since $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$, the function is one-one.$II$. For onto check: The range of $f(x) = 3^{-x}$ is $(0, \infty)$, which is a subset of the codomain $R$. Since the range $
eq$ codomain, the function is not onto.$III$. For decreasing check: Differentiating $f(x)$ with respect to $x$, we get $f'(x) = -3^{-x} \ln 3$. Since $3^{-x} > 0$ and $\ln 3 > 0$, $f'(x) Therefore, statements $I$ and $III$ are true.

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

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