Q43 / 160 · AP EAMCET 2004 · Mathematics

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

  • Correct answer
    A
    Only I, II
  • B
    Only II,III
  • C
    Only I,III
  • D
    I,II,III

Solution

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.

Hence, the correct answer is option: 'A'.

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