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

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, only I, II, only II, III, only I, III, I, II, III

MARKS App · Accepted Answer

Since, f : R → 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 ⇒ 3 - x 1 = 3 - x 2 ⇒ 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 d y d x = - 3 - x log ⁡ 3 0 for every value of x . ∴ It is decreasing function. ∴ Statement I and III are true.

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