AP EAMCET · Maths · Functions

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

A Only I, II
B Only II,III
C Only I,III
D I,II,III

Verified Solution

Answer & Solution

Correct Answer

(A) Only I, II

Step-by-step Solution

Detailed explanation

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…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The function \(f: R \rightarrow R\) is defined by \(f(x)=3^{-x}\). Observe the following statements of itI. \(f\) is one-oneII. \(f\) is ontoIII. \(f\) is a decreasing functionOut of these, true statement are

Step-by-step Solution

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