AP EAMCET · Maths · Functions

Let \(a\gt1\) and \(0 \lt b \lt 1\). If \(f: \mathbf{R} \rightarrow[0,1]\) is defined by
\(f(x)=\left\{\begin{array}{l}a^x,-\infty \lt x \lt 0 \\ b^x, 0 \leq x \lt \infty\end{array}\right.\) then \(f(x)\) is

A A bijection
B One-one but not onto
C Onto but not one-one-
D Neither one-one nor onto

Verified Solution

Answer & Solution

Correct Answer

(D) Neither one-one nor onto

Step-by-step Solution

Detailed explanation

\(f(x)= \begin{cases}a^x, & -\infty \lt x \lt 0 \\ b^x, & 0 \leq x \lt \infty\end{cases}\) \(a\gt1\) and \(0 \lt b \lt 1\) So by the graph Clearly, every horizontal line cuts \(f(x)\) at 2 points. Also, for \(f(x)=0\) There is no \(x \in \mathbb{R}\) So, \(f(x)\) is neither…

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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Let \(a\gt1\) and \(0 \lt b \lt 1\). If \(f: \mathbf{R} \rightarrow[0,1]\) is defined by \(f(x)=\left\{\begin{array}{l}a^x,-\infty \lt x \lt 0 \\ b^x, 0 \leq x \lt \infty\end{array}\right.\) then \(f(x)\) is

Step-by-step Solution

Let \(a\gt1\) and \(0 \lt b \lt 1\). If \(f: \mathbf{R} \rightarrow[0,1]\) is defined by
\(f(x)=\left\{\begin{array}{l}a^x,-\infty \lt x \lt 0 \\ b^x, 0 \leq x \lt \infty\end{array}\right.\) then \(f(x)\) is