Let f: R- \ -1/2 \ rightarrow R and g: R- \ -5/2 \ rightarrow R be defined as f(x)=2 x+3/2 x+1 and g(x)= x +1/2 x+5 . Then the domain of the function fog is :

Mathematics · Functions

JEE Main 2024 — 27 January, Shift 2 — Question 6

Let $f: R-\left\{\frac{-1}{2}\right\} \rightarrow R$ and $g: R-\left\{\frac{-5}{2}\right\} \rightarrow R$ be defined as $f(x)=\frac{2 x+3}{2 x+1}$ and $g(x)=\frac{|x|+1}{2 x+5}$ . Then the domain of the function fog is :

Option A:
$\mathrm{R}-\left\{-\frac{5}{2}\right\}$
Correct
Option B:
R
Option C:
$\mathrm{R}-\left\{-\frac{7}{4}\right\}$
Option D:
$\mathrm{R}-\left\{-\frac{5}{2},-\frac{7}{4}\right\}$

Answer: A

Step-by-step solution

$\mathrm{f}(\mathrm{x})=\frac{2 \mathrm{x}+3}{2 \mathrm{x}+1} ; \mathrm{x} \neq-\frac{1}{2}$

$g(x)=\frac{|x|+1}{2 x+5}, x \neq-\frac{5}{2}$

Domain of $\mathrm{f}(\mathrm{g}(\mathrm{x}))$

$\mathrm{f}(\mathrm{g}(\mathrm{x}))=\frac{2 \mathrm{~g}(\mathrm{x})+3}{2 \mathrm{~g}(\mathrm{x})+1}$

$x \neq-\frac{5}{2}$ and $\frac{|x|+1}{2 x+5} \neq-\frac{1}{2}$

$x \in R-\left\{-\frac{5}{2}\right\}$ and $x \in R$

$\therefore$ Domain will be $\mathrm{R}-\left\{-\frac{5}{2}\right\}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 27 January, Shift 2
Subject: Mathematics
Chapter: Functions
Topic: Composite Functions