JEE MAIN - Mathematics (2024 - 27th January Evening Shift - No. 10)
Let $$f: \mathbf{R}-\left\{\frac{-1}{2}\right\} \rightarrow \mathbf{R}$$ and $$g: \mathbf{R}-\left\{\frac{-5}{2}\right\} \rightarrow \mathbf{R}$$ 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 :
$$\mathbf{R}-\left\{-\frac{7}{4}\right\}$$
$$\mathbf{R}$$
$$\mathbf{R}-\left\{-\frac{5}{2},-\frac{7}{4}\right\}$$
$$\mathbf{R}-\left\{-\frac{5}{2}\right\}$$
Explanation
$$\begin{aligned} & f(x)=\frac{2 x+3}{2 x+1} ; x \neq-\frac{1}{2} \\ & g(x)=\frac{|x|+1}{2 x+5}, x \neq-\frac{5}{2} \end{aligned}$$
Domain of $$f(g(x))$$
$$f(g(x))=\frac{2 g(x)+3}{2 g(x)+1}$$
$$x \neq-\frac{5}{2}$$ 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 $$\mathrm{R}-\left\{-\frac{5}{2}\right\}$$
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