Let [f: R-(-1)/(2) R] and [g: R-(-5)/(2) R] be defined as [f(x)=(2 x+3)/(2 x+1)] and [g(x)…

Question

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 :

Accepted Answer

$$\begin{aligned}
& f(x)=\frac{2 x+3}{2 x+1} ; x \neq-\frac{1}{2} \
& g(x)=\frac{|x|+1}{2 x+5}, x \neq-\frac{5}{2}
\end{aligned}$$

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 $$\mathrm{R}-\left\{-\frac{5}{2}\right\}$$

Solutions · 1