Question

A $\subseteq \mathbb{R}$ and $f:A \to \mathbb{R}$ is a function defined by $f(x)=\dfrac{x-1}{x+1}$. If the domain of the function $f(2x)$ is $B$, then $A \cap B =$

• A. $\mathbb{R}\setminus\{-1,-2\}$
• B. $\mathbb{R}\setminus\left\{-1,-\dfrac12\right\}$
• C. $\mathbb{R}\setminus\left\{-1,\dfrac13\right\}$
• D. $\mathbb{R}\setminus\{-1,1\}$

Hint:

For rational functions, exclude every value that makes the denominator equal to zero.

Answer 1

The Correct Option is B

Step 1: Concept
The domain of a rational function excludes values that make the denominator zero.

Step 2: Meaning
For $f(x)=\dfrac{x-1}{x+1}$, the denominator vanishes at $x=-1$, so $A=\mathbb{R}\setminus\{-1\}$.

Step 3: Analysis
For $f(2x)$, \[ f(2x)=\frac{2x-1}{2x+1}. \] The denominator is zero when $2x+1=0$, giving $x=-\dfrac12$. Hence \[ B=\mathbb{R}\setminus\left\{-\frac12\right\}. \]

Step 4: Conclusion
Therefore, \[ A\cap B=\mathbb{R}\setminus\left\{-1,-\frac12\right\}. \]

Final Answer: (B)