CAT 2025 Slot 2 QA Q4 Functions & Graphs

Question 4 of the Quantitative Ability (QA) section, CAT 2025 Slot 2, with CAT's official answer and a worked solution.

Let \( f ( x ) = \frac { x } { ( 2 x - 1 ) } \) and \( g ( x ) = \frac { x } { ( x - 1 ) } \). Then, the domain of the function \( h ( x ) = f ( g ( x ) ) + g ( f ( x ) ) \) is all real numbers except

A
\( - 1 , \frac { 1 } { 2 } \), and 1
B
\( \frac { 1 } { 2 } , 1 \), and \( \frac { 3 } { 2 } \)
C
\( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \), and 1
D
\( \frac { 1 } { 2 } \), and 1

Answer: (A) - 1 , ( 1 )/( 2 ) , and 1 Functions & Graphs MCQ

Solution

\(f(g(x))\) needs \(x \ne 1\) (for \(g\)) and \(2g(x) - 1 \ne 0\), i.e. \(\frac{x}{x-1} \ne \frac12 \Rightarrow 2x \ne x - 1 \Rightarrow x \ne -1\).
\(g(f(x))\) needs \(x \ne \frac12\) (for \(f\)) and \(f(x) \ne 1\), i.e. \(x \ne 2x - 1 \Rightarrow x \ne 1\).
Excluded values: \(-1, \frac12, 1\).

Shortcut. Remember to exclude both the inner function's own poles and the points where the inner value hits the outer pole.