Functions Question 4
Question 4 - 2024 (27 Jan Shift 2)
Let $\mathrm{f}: \mathrm{R}-\left{\frac{-1}{2}\right} \rightarrow \mathrm{R}$ and $\mathrm{g}: \mathrm{R}-\left{\frac{-5}{2}\right} \rightarrow \mathrm{R}$ 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 :
(1) $\mathrm{R}-\left{-\frac{5}{2}\right}$
(2) R
(3) $\mathrm{R}-\left{-\frac{7}{4}\right}$
(4) $\mathrm{R}-\left{-\frac{5}{2},-\frac{7}{4}\right}$
Show Answer
Answer (1)
Solution
$\mathrm{f}(\mathrm{x})=\frac{2 \mathrm{x}+3}{2 \mathrm{x}+1} ; \mathrm{x} \neq-\frac{1}{2}$
$\mathrm{g}(\mathrm{x})=\frac{|\mathrm{x}|+1}{2 \mathrm{x}+5}, \mathrm{x} \neq-\frac{5}{2}$
Domain of $f(g(x))$
$f(g(x))=\frac{2 g(x)+3}{2 g(x)+1}$
$\mathbf{x} \neq-\frac{5}{2}$ and $\frac{|\mathrm{x}|+1}{2 \mathrm{x}+5} \neq-\frac{1}{2}$
$x \in R-\left{-\frac{5}{2}\right}$ and $x \in R$
$\therefore$ Domain will be $\mathrm{R}-\left{-\frac{5}{2}\right}$
