Sets, Relations & Functions
Domain of Composition f(g(x))
nta_pyq_2024_jan
Grade 11
Question:
Let $f:\mathbb{R}-\left\{-\dfrac{1}{2}\right\}\to\mathbb{R}$ and $g:\mathbb{R}-\left\{-\dfrac{5}{2}\right\}\to\mathbb{R}$ be defined as $f(x)=\dfrac{2x+3}{2x+1}$ and $g(x)=\dfrac{|x|+1}{2x+5}$. Then the domain of the function $f\circ g$ is:
$\mathbb{R}-\left\{-\dfrac{5}{2}\right\}$
$\mathbb{R}$
$\mathbb{R}-\left\{-\dfrac{7}{4}\right\}$
$\mathbb{R}-\left\{-\dfrac{5}{2},-\dfrac{7}{4}\right\}$
Step-by-Step Solution
Key Concept: Domain of $f(g(x))$: need $x\neq-5/2$ (domain of $g$) AND $g(x)\neq-1/2$ (domain of $f$). Solve $g(x)=-1/2$ and check if solutions are real.
$g(x)=-1/2$ has no solution. Domain of $f(g(x))=$ domain of $g=\mathbb{R}-\{-5/2\}$.
Correct Answer: 1